var Component=(()=>{var p=Object.create;var m=Object.defineProperty;var N=Object.getOwnPropertyDescriptor;var o=Object.getOwnPropertyNames;var g=Object.getPrototypeOf,y=Object.prototype.hasOwnProperty;var v=(a,s)=>()=>(s||a((s={exports:{}}).exports,s),s.exports),u=(a,s)=>{for(var l in s)m(a,l,{get:s[l],enumerable:!0})},c=(a,s,l,i)=>{if(s&&typeof s=="object"||typeof s=="function")for(let n of o(s))!y.call(a,n)&&n!==l&&m(a,n,{get:()=>s[n],enumerable:!(i=N(s,n))||i.enumerable});return a};var x=(a,s,l)=>(l=a!=null?p(g(a)):{},c(s||!a||!a.__esModule?m(l,"default",{value:a,enumerable:!0}):l,a)),w=a=>c(m({},"__esModule",{value:!0}),a);var r=v((M,t)=>{t.exports=_jsx_runtime});var R={};u(R,{default:()=>d,frontmatter:()=>b});var e=x(r()),b={title:"Additional Mathematics Formula Book",date:"2026-05-17",author:"Matthew Williams",subject:"Add Math",tags:["Section 1","Section 2","Section 3","Section 4","Reference","Paper 01","Paper 02"],description:"All key formulas, identities, rules, and results organised by syllabus section and topic for CSEC Additional Mathematics."};function h(a){let s={annotation:"annotation",h1:"h1",h2:"h2",h3:"h3",li:"li",math:"math",mfrac:"mfrac",mi:"mi",mn:"mn",mo:"mo",mover:"mover",mroot:"mroot",mrow:"mrow",mspace:"mspace",msqrt:"msqrt",mstyle:"mstyle",msub:"msub",msubsup:"msubsup",msup:"msup",mtable:"mtable",mtd:"mtd",mtext:"mtext",mtr:"mtr",munderover:"munderover",p:"p",path:"path",semantics:"semantics",span:"span",strong:"strong",svg:"svg",table:"table",tbody:"tbody",td:"td",th:"th",thead:"thead",tr:"tr",ul:"ul",...a.components};return(0,e.jsxs)(e.Fragment,{children:[(0,e.jsx)(s.h1,{id:"additional-mathematics-formula-book",children:"Additional Mathematics Formula Book"}),` `,(0,e.jsx)(s.h2,{id:"section-1-algebra-sequences-and-series",children:"Section 1: Algebra, Sequences and Series"}),` `,(0,e.jsx)(s.h3,{id:"a-algebra",children:"A. Algebra"}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Remainder Theorem:"})," When ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f(x)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})," is divided by ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(x - a)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]}),", the remainder is ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f(a)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]}),"."]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Factor Theorem:"})," ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(x - a)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})," is a factor of ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f(x)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})," if and only if ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f(a) = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]}),"."]}),` `,(0,e.jsx)(s.h3,{id:"b-quadratics",children:"B. Quadratics"}),` `,(0,e.jsxs)(s.p,{children:["For ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"c"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"ax^2 + bx + c = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7778em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]}),":"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mo,{children:"\xB1"}),(0,e.jsx)(s.msqrt,{children:(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"4"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"c"})]})})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"a"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.3845em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.0395em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mtight",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"b"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\xB1"}),(0,e.jsx)(s.span,{className:"mord sqrt mtight",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9221em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord mtight",style:{paddingLeft:"0.833em"},children:[(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"b"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7463em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.786em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mtight",children:"4"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"c"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8821em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail mtight",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1179em"},children:(0,e.jsx)(s.span,{})})})]})})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Vertex form"})," ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"h"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"k"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a(x + h)^2 + k"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0641em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"h"}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"})]})]})]}),", vertex at ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"h"}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mi,{children:"k"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(-h,\\, k)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"h"}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]}),":"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"h"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"a"})]})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"k"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"c"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"h"}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"h = \\frac{b}{2a} \\qquad k = c - ah^2"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"h"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2251em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8801em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mtight",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"b"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8141em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"h"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Discriminant"})," ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{mathvariant:"normal",children:"\u0394"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"4"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"c"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\Delta = b^2 - 4ac"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u0394"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"4"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"})]})]})]}),":"]}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{mathvariant:"normal",children:"\u0394"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\Delta"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u0394"})]})})]})}),(0,e.jsx)(s.th,{children:"Nature of roots"})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{mathvariant:"normal",children:"\u0394"}),(0,e.jsx)(s.mo,{children:">"}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\Delta > 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7224em",verticalAlign:"-0.0391em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u0394"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:">"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]})}),(0,e.jsx)(s.td,{children:"Two distinct real roots"})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{mathvariant:"normal",children:"\u0394"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\Delta = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u0394"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]})}),(0,e.jsx)(s.td,{children:"One repeated real root"})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{mathvariant:"normal",children:"\u0394"}),(0,e.jsx)(s.mo,{children:"<"}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\Delta < 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7224em",verticalAlign:"-0.0391em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u0394"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"<"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]})}),(0,e.jsx)(s.td,{children:"No real roots"})]})]})]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Vieta's formulas"})," for roots ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"\u03B1"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\alpha"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"})]})})]})," and ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"\u03B2"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\beta"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"})]})})]})," of ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"c"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"ax^2 + bx + c = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7778em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]}),":"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"\u03B2"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mi,{children:"\u03B2"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"c"}),(0,e.jsx)(s.mi,{children:"a"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\alpha + \\beta = -\\frac{b}{a} \\qquad \\alpha\\beta = \\frac{c}{a}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2251em",verticalAlign:"-0.345em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u2212"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8801em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"b"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0404em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.6954em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"c"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:"Expression"}),(0,e.jsx)(s.th,{children:"How to evaluate"})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"\u03B2"}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\alpha^2 + \\beta^2"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0085em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"\u03B2"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mi,{children:"\u03B2"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(\\alpha + \\beta)^2 - 2\\alpha\\beta"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0641em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"})]})]})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"\u03B2"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(\\alpha - \\beta)^2"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0641em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"\u03B2"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"4"}),(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mi,{children:"\u03B2"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(\\alpha + \\beta)^2 - 4\\alpha\\beta"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0641em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord",children:"4"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"})]})]})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mi,{children:"\u03B1"})]})}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mi,{children:"\u03B2"})]})})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{1}{\\alpha} + \\dfrac{1}{\\beta}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3214em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"1"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.2019em",verticalAlign:"-0.8804em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3214em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"1"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8804em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"\u03B2"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mi,{children:"\u03B2"})]})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{\\alpha + \\beta}{\\alpha\\beta}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.2519em",verticalAlign:"-0.8804em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3714em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8804em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"\u03B2"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"\u03B2"}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\alpha^2\\beta + \\alpha\\beta^2"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0085em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0085em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mi,{children:"\u03B2"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"\u03B1"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"\u03B2"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\alpha\\beta(\\alpha + \\beta)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03B1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0528em"},children:"\u03B2"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})})]})]})]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Forming a new quadratic"})," from known sum ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"S"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"S"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0576em"},children:"S"})]})})]})," and product ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"P"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"P"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"})]})})]}),":"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"S"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"x^2 - Sx + P = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7667em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0576em"},children:"S"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:["Multiply through by a constant if needed so that ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsx)(s.mi,{children:"c"}),(0,e.jsx)(s.mo,{children:"\u2208"}),(0,e.jsx)(s.mi,{mathvariant:"double-struck",children:"Z"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a, b, c \\in \\mathbb{Z}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"\u2208"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6889em"}}),(0,e.jsx)(s.span,{className:"mord mathbb",children:"Z"})]})]})]}),"."]}),` `,(0,e.jsx)(s.h3,{id:"c-inequalities",children:"C. Inequalities"}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Quadratic inequalities:"})," find the critical values (roots), sketch the parabola, read the solution set from the graph."]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Rational inequalities"})," ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"g"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]})]})}),(0,e.jsx)(s.mo,{children:"\u2276"}),(0,e.jsx)(s.mi,{children:"k"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{f(x)}{g(x)} \\lessgtr k"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.363em",verticalAlign:"-0.936em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.427em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"g"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.936em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel amsrm",children:"\u2276"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"})]})]})]}),": rearrange to get zero on one side, find critical values (roots and excluded values), apply a sign diagram."]}),` `,(0,e.jsx)(s.h3,{id:"d-surds-indices-and-logarithms",children:"D. Surds, Indices, and Logarithms"}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Surd rules:"})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mi,{children:"a"})}),(0,e.jsx)(s.mo,{children:"\xD7"}),(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mi,{children:"b"})}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.msqrt,{children:(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"b"})]})}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mi,{children:"a"})}),(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mi,{children:"b"})})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.msqrt,{children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"b"})]})}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mi,{children:"a"})}),(0,e.jsx)(s.mo,{children:"\xD7"}),(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mi,{children:"a"})}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\sqrt{a} \\times \\sqrt{b} = \\sqrt{ab} \\qquad \\frac{\\sqrt{a}}{\\sqrt{b}} = \\sqrt{\\frac{a}{b}} \\qquad \\sqrt{a} \\times \\sqrt{a} = a"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.04em",verticalAlign:"-0.2397em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8003em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.7603em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2397em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\xD7"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.04em",verticalAlign:"-0.1078em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9322em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8922em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1078em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.576em",verticalAlign:"-0.538em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9322em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"ab"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8922em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1078em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.038em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.5335em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord sqrt mtight",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9378em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord mtight",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"b"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8978em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail mtight",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1022em"},children:(0,e.jsx)(s.span,{})})})]})})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.4739em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord sqrt mtight",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8059em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord mtight",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.7659em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail mtight",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2341em"},children:(0,e.jsx)(s.span,{})})})]})})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.538em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.24em",verticalAlign:"-0.3798em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8602em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3.2em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.2em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"1em"},children:(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.6954em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"b"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8202em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.2em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"1.02em",height:"1.28em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 c4.7,-7.3,11,-11,19,-11 H40000v40H1012.3 s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232 c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z M1001 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3798em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8003em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.7603em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2397em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\xD7"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.04em",verticalAlign:"-0.2397em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8003em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.7603em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2397em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:["Rationalise using the conjugate: ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mi,{children:"a"})}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mi,{children:"b"})}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mi,{children:"a"})}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mi,{children:"b"})}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"b"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(\\sqrt{a} + \\sqrt{b})(\\sqrt{a} - \\sqrt{b}) = a - b"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0503em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8003em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.7603em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2397em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1822em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9322em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8922em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1078em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8003em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.7603em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2397em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1822em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9322em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8922em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1078em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"})]})]})]}),"."]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Laws of indices:"})}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:"Law"}),(0,e.jsx)(s.th,{children:"Rule"})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"m"})]}),(0,e.jsx)(s.mo,{children:"\xD7"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"n"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^m \\times a^n"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7477em",verticalAlign:"-0.0833em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\xD7"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6644em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"n"})]})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^{m+n}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7713em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7713em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"+"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})]})})]})})})})})]})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"m"})]}),(0,e.jsx)(s.mo,{children:"\xF7"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"n"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^m \\div a^n"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7477em",verticalAlign:"-0.0833em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\xF7"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6644em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"n"})]})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^{m-n}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7713em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7713em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})]})})]})})})})})]})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"m"})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mi,{children:"n"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(a^m)^n"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"})})]})})})})})]}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mi,{children:"n"})]})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^{mn}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6644em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"mn"})})})]})})})})})]})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mn,{children:"0"})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^0"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8141em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"0"})})]})})})})})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mn,{children:"1"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"1"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"n"})]})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^{-n}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7713em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7713em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})]})})]})})})})})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"n"})]})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{1}{a^n}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3214em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.5904em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.989em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"1"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mi,{mathvariant:"normal",children:"/"}),(0,e.jsx)(s.mi,{children:"n"})]})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^{m/n}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.888em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.888em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"}),(0,e.jsx)(s.span,{className:"mord mtight",children:"/"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})]})})]})})})})})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.mroot,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"m"})]}),(0,e.jsx)(s.mi,{children:"n"})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\sqrt[n]{a^m}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.04em",verticalAlign:"-0.1598em"}}),(0,e.jsxs)(s.span,{className:"mord sqrt",children:[(0,e.jsx)(s.span,{className:"root",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6475em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.9322em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})})]})})})})}),(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8802em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.5904em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.989em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"})})]})})})})})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8402em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1598em"},children:(0,e.jsx)(s.span,{})})})]})]})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mi,{children:"m"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(ab)^m"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"ab"}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"})})]})})})})})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"m"})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mi,{children:"m"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^m b^m"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"})})]})})})})})]}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"})})]})})})})})]})]})})]})})]})]})]}),` `,(0,e.jsxs)(s.p,{children:["If ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"m"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"n"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^m = a^n"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6644em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"m"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6644em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]})]})]})]})," and ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{mathvariant:"normal",children:"\u2260"}),(0,e.jsx)(s.mn,{children:"0"}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a \\neq 0,\\, 1"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsxs)(s.span,{className:"mrel",children:[(0,e.jsx)(s.span,{className:"mrel",children:(0,e.jsx)(s.span,{className:"mord vbox",children:(0,e.jsx)(s.span,{className:"thinbox",children:(0,e.jsxs)(s.span,{className:"rlap",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"inner",children:(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mrel",children:"\uE020"})})}),(0,e.jsx)(s.span,{className:"fix"})]})})})}),(0,e.jsx)(s.span,{className:"mspace nobreak"}),(0,e.jsx)(s.span,{className:"mrel",children:"="})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8389em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})]})]}),", then ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"m = n"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"m"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"n"})]})]})]}),"."]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Logarithm definition:"})," ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mtext,{children:"\u2005\u200A"}),(0,e.jsx)(s.mo,{children:"\u27FA"}),(0,e.jsx)(s.mtext,{children:"\u2005\u200A"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mspace,{width:"1em"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{children:">"}),(0,e.jsx)(s.mn,{children:"0"}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsx)(s.mtext,{children:"\u2005\u200A"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{mathvariant:"normal",children:"\u2260"}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsx)(s.mtext,{children:"\u2005\u200A"}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mo,{children:">"}),(0,e.jsx)(s.mn,{children:"0"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^x = b \\iff \\log_a b = x \\quad (a > 0,\\; a \\neq 1,\\; b > 0)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6644em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7184em",verticalAlign:"-0.024em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"\u27FA"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.9386em",verticalAlign:"-0.2441em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"1em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:">"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsxs)(s.span,{className:"mrel",children:[(0,e.jsx)(s.span,{className:"mrel",children:(0,e.jsx)(s.span,{className:"mord vbox",children:(0,e.jsx)(s.span,{className:"thinbox",children:(0,e.jsxs)(s.span,{className:"rlap",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"inner",children:(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mrel",children:"\uE020"})})}),(0,e.jsx)(s.span,{className:"fix"})]})})})}),(0,e.jsx)(s.span,{className:"mspace nobreak"}),(0,e.jsx)(s.span,{className:"mrel",children:"="})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:">"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})]}),` `,(0,e.jsxs)(s.p,{children:["Key values: ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log_a 1 = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.9386em",verticalAlign:"-0.2441em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]}),", ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log_a a = 1"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.9386em",verticalAlign:"-0.2441em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})]})]}),"."]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Laws of logarithms:"})}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:"Law"}),(0,e.jsx)(s.th,{children:"Statement"})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:"Product"}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mi,{children:"y"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log_a(xy) = \\log_a x + \\log_a y"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.9386em",verticalAlign:"-0.2441em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.9386em",verticalAlign:"-0.2441em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"})]})]})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:"Quotient"}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mtext,{children:"\u2009\u2063"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{fence:"true",children:"("}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mi,{children:"y"})]})}),(0,e.jsx)(s.mo,{fence:"true",children:")"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mi,{children:"y"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log_a\\!\\left(\\dfrac{x}{y}\\right) = \\log_a x - \\log_a y"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.4em",verticalAlign:"-0.95em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"-0.1667em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"minner",children:[(0,e.jsx)(s.span,{className:"mopen delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size3",children:"("})}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.1076em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8804em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mclose delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size3",children:")"})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.9386em",verticalAlign:"-0.2441em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.9386em",verticalAlign:"-0.2441em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"})]})]})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:"Power"}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mi,{children:"x"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log_a(x^n) = n\\log_a x"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.9386em",verticalAlign:"-0.2441em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"n"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.0573em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.4559em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2441em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})]})]})})]})]})]}),` `,(0,e.jsxs)(s.p,{children:["To solve ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"b"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a^x = b"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6644em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"})]})]})]}),": apply ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]})]})})]})," to both sides and use the power law: ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"b"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"a"})]})]})})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"x = \\dfrac{\\log b}{\\log a}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.2519em",verticalAlign:"-0.8804em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3714em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8804em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]}),"."]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Linearisation:"})}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:"Relationship"}),(0,e.jsx)(s.th,{children:"Linearised form"}),(0,e.jsx)(s.th,{children:"Plot"}),(0,e.jsx)(s.th,{children:"Gradient"}),(0,e.jsx)(s.th,{children:"Intercept"})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mi,{children:"x"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"y = ab^x"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})})]})})})})})]})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log y = x\\log b + \\log a"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})]})]})]})}),(0,e.jsxs)(s.td,{children:[(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"y"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log y"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"})]})})]})," vs ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"x"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"x"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})})]})]}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"b"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log b"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log a"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mi,{children:"n"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"y = ax^n"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6644em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log y = n\\log x + \\log a"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"n"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})]})]})]})}),(0,e.jsxs)(s.td,{children:[(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"y"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log y"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"})]})})]})," vs ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"x"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log x"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})})]})]}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"n"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"n"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"log"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\log a"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mop",children:["lo",(0,e.jsx)(s.span,{style:{marginRight:"0.0139em"},children:"g"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})]})})]})})]})]})]}),` `,(0,e.jsx)(s.h3,{id:"e-sequences-and-series",children:"E. Sequences and Series"}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Sigma notation:"})," ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:[(0,e.jsxs)(s.munderover,{children:[(0,e.jsx)(s.mo,{children:"\u2211"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"U"}),(0,e.jsx)(s.mi,{children:"r"})]})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\displaystyle\\sum_{r=1}^{n} U_r"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.9185em",verticalAlign:"-1.2671em"}}),(0,e.jsx)(s.span,{className:"mop op-limits",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.6514em"},children:[(0,e.jsxs)(s.span,{style:{top:"-1.8829em",marginLeft:"0em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.05em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"mrel mtight",children:"="}),(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.05em"}}),(0,e.jsx)(s.span,{children:(0,e.jsx)(s.span,{className:"mop op-symbol large-op",children:"\u2211"})})]}),(0,e.jsxs)(s.span,{style:{top:"-4.3em",marginLeft:"0em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.05em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"1.2671em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.109em"},children:"U"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1514em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0278em"},children:"r"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]})]})})]})," means the sum of ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"U"}),(0,e.jsx)(s.mi,{children:"r"})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"U_r"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.109em"},children:"U"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1514em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0278em"},children:"r"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]})]})})]})," from ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"r = 1"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})]})]})," to ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"r = n"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"n"})]})]})]}),"."]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Arithmetic sequences"})," (",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"a"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})]})})]})," = first term, ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"d"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"d"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"})]})})]})," = common difference, ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"l"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"l"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0197em"},children:"l"})]})})]})," = last term, ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"n"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"n"})]})})]})," terms):"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"T"}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"S"}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"["}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"]"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"l"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"T_n = a + (n-1)d \\qquad S_n = \\frac{n}{2}[2a + (n-1)d] = \\frac{n}{2}(a + l)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"T"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1514em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.1389em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"n"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0576em"},children:"S"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1514em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.0576em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.095em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.6954em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mopen",children:"["}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"n"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mclose",children:"]"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.095em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.6954em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0197em"},children:"l"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Geometric sequences"})," (",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"a"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})]})})]})," = first term, ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"r"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"r"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"})]})})]})," = common ratio):"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"T"}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"1"})]})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"T"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mn,{children:"1"})]})]}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"T"}),(0,e.jsx)(s.mi,{children:"n"})]})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"S"}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"r"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"T_n = ar^{n-1} \\qquad r = \\frac{T_{n+1}}{T_n} \\qquad S_n = \\frac{a(1 - r^n)}{1 - r}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"T"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1514em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.1389em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8141em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})]})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.3752em",verticalAlign:"-0.4451em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9301em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.1389em"},children:"T"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1645em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"-0.1389em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.4518em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.1389em"},children:"T"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"-0.1389em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"+"}),(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})]})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2025em"},children:(0,e.jsx)(s.span,{})})})]})})]})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.4451em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0576em"},children:"S"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1514em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.0576em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.4133em",verticalAlign:"-0.4033em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.01em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mtight",children:"1"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0278em"},children:"r"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.485em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mopen mtight",children:"("}),(0,e.jsx)(s.span,{className:"mord mtight",children:"1"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7385em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.931em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mclose mtight",children:")"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.4033em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Sum to infinity"})," (converges when ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"}),(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"}),(0,e.jsx)(s.mo,{children:"<"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\lvert r \\rvert < 1"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"mclose",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"<"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})]})]})," only):"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"S"}),(0,e.jsx)(s.mi,{mathvariant:"normal",children:"\u221E"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"r"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"S_\\infty = \\frac{a}{1 - r}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0576em"},children:"S"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1514em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.0576em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u221E"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0987em",verticalAlign:"-0.4033em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.6954em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mtight",children:"1"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0278em"},children:"r"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.4033em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsx)(s.h2,{id:"section-2-coordinate-geometry-vectors-and-trigonometry",children:"Section 2: Coordinate Geometry, Vectors, and Trigonometry"}),` `,(0,e.jsx)(s.h3,{id:"a-coordinate-geometry",children:"A. Coordinate Geometry"}),` `,(0,e.jsxs)(s.p,{children:["For two points ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(x_1, y_1)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.0359em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})," and ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(x_2, y_2)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.0359em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]}),":"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"1"})]})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"1"})]})]})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.msqrt,{children:(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]})]})}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"M"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{fence:"true",children:"("}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsx)(s.mtext,{children:"\u2005\u200A"}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{fence:"true",children:")"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"m = \\frac{y_2 - y_1}{x_2 - x_1} \\qquad d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\qquad M = \\left(\\frac{x_1 + x_2}{2},\\; \\frac{y_1 + y_2}{2}\\right)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"m"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2995em",verticalAlign:"-0.4451em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8544em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"0em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"0em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.4461em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"-0.0359em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"-0.0359em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.4451em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.24em",verticalAlign:"-0.305em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.935em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3.2em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.2em"}}),(0,e.jsxs)(s.span,{className:"mord",style:{paddingLeft:"1em"},children:[(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7401em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.989em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.0359em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.0359em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7401em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.989em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})]}),(0,e.jsxs)(s.span,{style:{top:"-2.895em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.2em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"1.02em",height:"1.28em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 c4.7,-7.3,11,-11,19,-11 H40000v40H1012.3 s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232 c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z M1001 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.305em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.109em"},children:"M"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2044em",verticalAlign:"-0.35em"}}),(0,e.jsxs)(s.span,{className:"minner",children:[(0,e.jsx)(s.span,{className:"mopen delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size1",children:"("})}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8184em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.4101em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"0em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"+"}),(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"0em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8544em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.4461em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"-0.0359em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"+"}),(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"-0.0359em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mclose delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size1",children:")"})})]})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:["Parallel lines: equal gradients. Perpendicular lines: ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"m_1 m_2 = -1"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"m"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"m"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7278em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})]})]}),"."]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Equation of a line:"})," ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"y - y_1 = m(x - x_1)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7778em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.0359em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"m"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Circle equations:"})}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:"Form"}),(0,e.jsx)(s.th,{children:"Equation"}),(0,e.jsx)(s.th,{children:"Centre"}),(0,e.jsx)(s.th,{children:"Radius"})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:"Standard"}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(x - a)^2 + (y - b)^2 = r^2"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0641em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0641em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8141em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(a, b)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"r"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"r"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:"General"}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"g"}),(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"c"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"x^2 + y^2 + 2fx + 2gy + c = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0085em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8389em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"g"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"g"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(-f, -g)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"g"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.msqrt,{children:(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"g"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"c"})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\sqrt{f^2 + g^2 - c}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.24em",verticalAlign:"-0.2822em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9578em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3.2em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.2em"}}),(0,e.jsxs)(s.span,{className:"mord",style:{paddingLeft:"1em"},children:[(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7401em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.989em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"g"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7401em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.989em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-2.9178em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.2em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"1.02em",height:"1.28em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 c4.7,-7.3,11,-11,19,-11 H40000v40H1012.3 s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232 c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z M1001 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2822em"},children:(0,e.jsx)(s.span,{})})})]})})]})})]})})]})]})]}),` `,(0,e.jsx)(s.p,{children:"The tangent at a point on a circle is perpendicular to the radius at that point."}),` `,(0,e.jsx)(s.h3,{id:"b-vectors",children:"B. Vectors"}),` `,(0,e.jsxs)(s.p,{children:["For ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{mathvariant:"bold",children:"v"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{fence:"true",children:"("}),(0,e.jsxs)(s.mtable,{rowspacing:"0.16em",columnalign:"center",columnspacing:"1em",children:[(0,e.jsx)(s.mtr,{children:(0,e.jsx)(s.mtd,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,e.jsx)(s.mi,{children:"x"})})})}),(0,e.jsx)(s.mtr,{children:(0,e.jsx)(s.mtd,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,e.jsx)(s.mi,{children:"y"})})})})]}),(0,e.jsx)(s.mo,{fence:"true",children:")"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\mathbf{v} = \\begin{pmatrix} x \\\\ y \\end{pmatrix}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4444em"}}),(0,e.jsx)(s.span,{className:"mord mathbf",style:{marginRight:"0.016em"},children:"v"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.4em",verticalAlign:"-0.95em"}}),(0,e.jsxs)(s.span,{className:"minner",children:[(0,e.jsx)(s.span,{className:"mopen delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size3",children:"("})}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mtable",children:(0,e.jsx)(s.span,{className:"col-align-c",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.45em"},children:[(0,e.jsxs)(s.span,{style:{top:"-3.61em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.41em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.95em"},children:(0,e.jsx)(s.span,{})})})]})})})}),(0,e.jsx)(s.span,{className:"mclose delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size3",children:")"})})]})]})]})]}),":"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"}),(0,e.jsx)(s.mi,{mathvariant:"bold",children:"v"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.msqrt,{children:(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"2"})]})]})}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsxs)(s.mover,{accent:"true",children:[(0,e.jsx)(s.mi,{mathvariant:"bold",children:"v"}),(0,e.jsx)(s.mo,{children:"^"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{mathvariant:"bold",children:"v"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"}),(0,e.jsx)(s.mi,{mathvariant:"bold",children:"v"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\lvert\\mathbf{v}\\rvert = \\sqrt{x^2 + y^2} \\qquad \\hat{\\mathbf{v}} = \\frac{\\mathbf{v}}{\\lvert\\mathbf{v}\\rvert}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mord mathbf",style:{marginRight:"0.016em"},children:"v"}),(0,e.jsx)(s.span,{className:"mclose",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.24em",verticalAlign:"-0.2822em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9578em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3.2em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.2em"}}),(0,e.jsxs)(s.span,{className:"mord",style:{paddingLeft:"1em"},children:[(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7401em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.989em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7401em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.989em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})]}),(0,e.jsxs)(s.span,{style:{top:"-2.9178em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.2em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"1.02em",height:"1.28em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 c4.7,-7.3,11,-11,19,-11 H40000v40H1012.3 s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232 c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z M1001 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2822em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord accent",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.7079em"},children:[(0,e.jsxs)(s.span,{style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord mathbf",style:{marginRight:"0.016em"},children:"v"})]}),(0,e.jsxs)(s.span,{style:{top:"-3.0134em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"accent-body",style:{left:"-0.25em"},children:(0,e.jsx)(s.span,{className:"mord",children:"^"})})]})]})})})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2251em",verticalAlign:"-0.52em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.7051em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mopen mtight",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mord mathbf mtight",style:{marginRight:"0.016em"},children:"v"}),(0,e.jsx)(s.span,{className:"mclose mtight",children:"\u2223"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathbf mtight",style:{marginRight:"0.016em"},children:"v"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.52em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Dot product:"})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{fence:"true",children:"("}),(0,e.jsxs)(s.mtable,{rowspacing:"0.16em",columnalign:"center",columnspacing:"1em",children:[(0,e.jsx)(s.mtr,{children:(0,e.jsx)(s.mtd,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,e.jsx)(s.mi,{children:"a"})})})}),(0,e.jsx)(s.mtr,{children:(0,e.jsx)(s.mtd,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,e.jsx)(s.mi,{children:"b"})})})})]}),(0,e.jsx)(s.mo,{fence:"true",children:")"})]}),(0,e.jsx)(s.mo,{children:"\u22C5"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{fence:"true",children:"("}),(0,e.jsxs)(s.mtable,{rowspacing:"0.16em",columnalign:"center",columnspacing:"1em",children:[(0,e.jsx)(s.mtr,{children:(0,e.jsx)(s.mtd,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,e.jsx)(s.mi,{children:"c"})})})}),(0,e.jsx)(s.mtr,{children:(0,e.jsx)(s.mtd,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,e.jsx)(s.mi,{children:"d"})})})})]}),(0,e.jsx)(s.mo,{fence:"true",children:")"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"c"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mi,{children:"d"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\begin{pmatrix} a \\\\ b \\end{pmatrix} \\cdot \\begin{pmatrix} c \\\\ d \\end{pmatrix} = ac + bd"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.4em",verticalAlign:"-0.95em"}}),(0,e.jsxs)(s.span,{className:"minner",children:[(0,e.jsx)(s.span,{className:"mopen delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size3",children:"("})}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mtable",children:(0,e.jsx)(s.span,{className:"col-align-c",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.45em"},children:[(0,e.jsxs)(s.span,{style:{top:"-3.61em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.41em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.95em"},children:(0,e.jsx)(s.span,{})})})]})})})}),(0,e.jsx)(s.span,{className:"mclose delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size3",children:")"})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u22C5"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.4em",verticalAlign:"-0.95em"}}),(0,e.jsxs)(s.span,{className:"minner",children:[(0,e.jsx)(s.span,{className:"mopen delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size3",children:"("})}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mtable",children:(0,e.jsx)(s.span,{className:"col-align-c",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.45em"},children:[(0,e.jsxs)(s.span,{style:{top:"-3.61em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.41em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.95em"},children:(0,e.jsx)(s.span,{})})})]})})})}),(0,e.jsx)(s.span,{className:"mclose delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size3",children:")"})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Angle between vectors:"})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{mathvariant:"bold",children:"a"}),(0,e.jsx)(s.mo,{children:"\u22C5"}),(0,e.jsx)(s.mi,{mathvariant:"bold",children:"b"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"}),(0,e.jsx)(s.mi,{mathvariant:"bold",children:"a"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"}),(0,e.jsx)(s.mi,{mathvariant:"bold",children:"b"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\cos\\theta = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\lvert\\mathbf{a}\\rvert\\lvert\\mathbf{b}\\rvert}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.4001em",verticalAlign:"-0.52em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8801em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mopen mtight",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mord mathbf mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mclose mtight",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mopen mtight",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mord mathbf mtight",children:"b"}),(0,e.jsx)(s.span,{className:"mclose mtight",children:"\u2223"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathbf mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u22C5"}),(0,e.jsx)(s.span,{className:"mord mathbf mtight",children:"b"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.52em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsxs)(s.ul,{children:[` `,(0,e.jsxs)(s.li,{children:[(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{mathvariant:"bold",children:"a"}),(0,e.jsx)(s.mo,{children:"\u22C5"}),(0,e.jsx)(s.mi,{mathvariant:"bold",children:"b"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\mathbf{a} \\cdot \\mathbf{b} = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4445em"}}),(0,e.jsx)(s.span,{className:"mord mathbf",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u22C5"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathbf",children:"b"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]}),": vectors are perpendicular."]}),` `,(0,e.jsxs)(s.li,{children:[(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{mathvariant:"bold",children:"b"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"k"}),(0,e.jsx)(s.mi,{mathvariant:"bold",children:"a"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\mathbf{b} = k\\mathbf{a}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathbf",children:"b"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"}),(0,e.jsx)(s.span,{className:"mord mathbf",children:"a"})]})]})]})," for some scalar ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"k"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"k"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"})]})})]}),": vectors are parallel."]}),` `,(0,e.jsxs)(s.li,{children:["Three points ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"A"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"A"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"})]})})]}),", ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"B"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"B"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"})]})})]}),", ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"C"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"C"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0715em"},children:"C"})]})})]})," are collinear if ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.mover,{accent:"true",children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mi,{children:"B"})]}),(0,e.jsx)(s.mo,{stretchy:"true",children:"\u2192"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"k"}),(0,e.jsxs)(s.mover,{accent:"true",children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mi,{children:"C"})]}),(0,e.jsx)(s.mo,{stretchy:"true",children:"\u2192"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\overrightarrow{AB} = k\\overrightarrow{AC}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2053em"}}),(0,e.jsx)(s.span,{className:"mord accent",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.2053em"},children:[(0,e.jsxs)(s.span,{style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"})]})]}),(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3.6833em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{height:"0.522em",minWidth:"0.888em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"0.522em",viewBox:"0 0 400000 522",preserveAspectRatio:"xMaxYMin slice",children:(0,e.jsx)(s.path,{d:`M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z`})})})]})]})})})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2053em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"}),(0,e.jsx)(s.span,{className:"mord accent",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.2053em"},children:[(0,e.jsxs)(s.span,{style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0715em"},children:"C"})]})]}),(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3.6833em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{height:"0.522em",minWidth:"0.888em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"0.522em",viewBox:"0 0 400000 522",preserveAspectRatio:"xMaxYMin slice",children:(0,e.jsx)(s.path,{d:`M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z`})})})]})]})})})})]})]})]}),"."]}),` `]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Direction of a vector:"})," ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"arctan"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mtext,{children:"\u2009\u2063"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{fence:"true",children:"("}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mi,{children:"x"})]})}),(0,e.jsx)(s.mo,{fence:"true",children:")"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\theta = \\arctan\\!\\left(\\dfrac{y}{x}\\right)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.836em",verticalAlign:"-0.686em"}}),(0,e.jsx)(s.span,{className:"mop",children:"arctan"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"-0.1667em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"minner",children:[(0,e.jsx)(s.span,{className:"mopen delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size2",children:"("})}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.1076em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mclose delimcenter",style:{top:"0em"},children:(0,e.jsx)(s.span,{className:"delimsizing size2",children:")"})})]})]})]})]}),", then adjust for the correct quadrant (add ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mn,{children:"180"}),(0,e.jsx)(s.mo,{children:"\u2218"})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"180^\\circ"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6741em"}}),(0,e.jsx)(s.span,{className:"mord",children:"18"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord",children:"0"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6741em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2218"})})]})})})})})]})]})})]})," in Q2 or Q3)."]}),` `,(0,e.jsx)(s.h3,{id:"c-trigonometry",children:"C. Trigonometry"}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Conversion:"})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mo,{children:"\u2218"})]}),(0,e.jsx)(s.mo,{children:"\xD7"}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"\u03C0"}),(0,e.jsx)(s.mn,{children:"180"})]}),(0,e.jsx)(s.mo,{children:"\u2192"}),(0,e.jsx)(s.mtext,{children:"rad"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mtext,{children:"\u2005\u200A"}),(0,e.jsx)(s.mtext,{children:"rad"}),(0,e.jsx)(s.mo,{children:"\xD7"}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"180"}),(0,e.jsx)(s.mi,{children:"\u03C0"})]}),(0,e.jsx)(s.mo,{children:"\u2192"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mrow,{}),(0,e.jsx)(s.mo,{children:"\u2218"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\theta^\\circ \\times \\frac{\\pi}{180} \\to \\text{rad} \\qquad \\theta\\;\\text{rad} \\times \\frac{180}{\\pi} \\to {}^\\circ"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7778em",verticalAlign:"-0.0833em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6741em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2218"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\xD7"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0404em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.6954em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"180"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"\u03C0"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"\u2192"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7778em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord text",children:(0,e.jsx)(s.span,{className:"mord",children:"rad"})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mord text",children:(0,e.jsx)(s.span,{className:"mord",children:"rad"})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\xD7"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1901em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8451em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"\u03C0"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"180"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"\u2192"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6741em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6741em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2218"})})]})})})})})]})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Arc and sector"})," (",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"\u03B8"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\theta"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"})]})})]})," in radians):"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"l"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mtext,{children:"sector"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mtext,{children:"segment"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"r"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"l = r\\theta \\qquad A_{\\text{sector}} = \\frac{1}{2}r^2\\theta \\qquad A_{\\text{segment}} = \\frac{1}{2}r^2(\\theta - \\sin\\theta)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0197em"},children:"l"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8444em",verticalAlign:"-0.15em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2806em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord text mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"sector"})})})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1901em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8451em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2806em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord text mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"segment"})})})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.2861em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1901em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8451em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"r"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Exact values:"})}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"\u03B8"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\theta"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"})]})})]})}),(0,e.jsx)(s.th,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mn,{children:"0"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"0"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})})]})}),(0,e.jsx)(s.th,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mn,{children:"30"}),(0,e.jsx)(s.mo,{children:"\u2218"})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"30^\\circ"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6741em"}}),(0,e.jsx)(s.span,{className:"mord",children:"3"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord",children:"0"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6741em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2218"})})]})})})})})]})]})})]})}),(0,e.jsx)(s.th,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mn,{children:"45"}),(0,e.jsx)(s.mo,{children:"\u2218"})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"45^\\circ"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6741em"}}),(0,e.jsx)(s.span,{className:"mord",children:"4"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord",children:"5"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6741em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2218"})})]})})})})})]})]})})]})}),(0,e.jsx)(s.th,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mn,{children:"60"}),(0,e.jsx)(s.mo,{children:"\u2218"})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"60^\\circ"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6741em"}}),(0,e.jsx)(s.span,{className:"mord",children:"6"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord",children:"0"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6741em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2218"})})]})})})})})]})]})})]})}),(0,e.jsx)(s.th,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mn,{children:"90"}),(0,e.jsx)(s.mo,{children:"\u2218"})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"90^\\circ"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6741em"}}),(0,e.jsx)(s.span,{className:"mord",children:"9"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord",children:"0"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6741em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2218"})})]})})})})})]})]})})]})})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"\u03B8"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\sin\\theta"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mn,{children:"0"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"0"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mn,{children:"2"})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{1}{2}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3214em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"2"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"1"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mn,{children:"2"})}),(0,e.jsx)(s.mn,{children:"2"})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{\\sqrt{2}}{2}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.2702em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.5842em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"2"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9072em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord",children:"2"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8672em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1328em"},children:(0,e.jsx)(s.span,{})})})]})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mn,{children:"3"})}),(0,e.jsx)(s.mn,{children:"2"})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{\\sqrt{3}}{2}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.2702em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.5842em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"2"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9072em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord",children:"3"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8672em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1328em"},children:(0,e.jsx)(s.span,{})})})]})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mn,{children:"1"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"1"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"\u03B8"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\cos\\theta"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mn,{children:"1"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"1"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mn,{children:"3"})}),(0,e.jsx)(s.mn,{children:"2"})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{\\sqrt{3}}{2}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.2702em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.5842em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"2"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9072em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord",children:"3"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8672em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1328em"},children:(0,e.jsx)(s.span,{})})})]})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mn,{children:"2"})}),(0,e.jsx)(s.mn,{children:"2"})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{\\sqrt{2}}{2}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.2702em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.5842em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"2"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9072em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord",children:"2"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8672em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1328em"},children:(0,e.jsx)(s.span,{})})})]})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mn,{children:"2"})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{1}{2}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3214em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"2"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"1"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mn,{children:"0"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"0"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"\u03B8"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\tan\\theta"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mop",children:"tan"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mn,{children:"0"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"0"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mn,{children:"3"})})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{1}{\\sqrt{3}}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.2514em",verticalAlign:"-0.93em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3214em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.2028em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9072em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord",children:"3"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8672em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1328em"},children:(0,e.jsx)(s.span,{})})})]})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"1"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.93em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mn,{children:"1"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"1"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.msqrt,{children:(0,e.jsx)(s.mn,{children:"3"})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\sqrt{3}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.04em",verticalAlign:"-0.1328em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9072em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"0.833em"},children:(0,e.jsx)(s.span,{className:"mord",children:"3"})})]}),(0,e.jsxs)(s.span,{style:{top:"-2.8672em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"0.853em",height:"1.08em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1328em"},children:(0,e.jsx)(s.span,{})})})]})})]})})]})}),(0,e.jsx)(s.td,{children:"undef."})]})]})]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"CAST rule:"})," All (Q1), Sine (Q2), Tangent (Q3), Cosine (Q4) are positive."]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Graph properties:"})}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:"Function"}),(0,e.jsx)(s.th,{children:"Period"}),(0,e.jsx)(s.th,{children:"Amplitude"})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"k"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"y = a\\sin(kx)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"\u03C0"})]}),(0,e.jsx)(s.mi,{children:"k"})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{2\\pi}{k}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3214em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"\u03C0"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\lvert a \\rvert"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mclose",children:"\u2223"})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"k"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"y = a\\cos(kx)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"\u03C0"})]}),(0,e.jsx)(s.mi,{children:"k"})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{2\\pi}{k}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3214em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"\u03C0"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"\u2223"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\lvert a \\rvert"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mclose",children:"\u2223"})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"k"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"y = a\\tan(kx)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"tan"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"\u03C0"}),(0,e.jsx)(s.mi,{children:"k"})]})})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{\\pi}{k}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.7936em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.1076em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0315em"},children:"k"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"\u03C0"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})}),(0,e.jsx)(s.td,{children:"\u2014"})]})]})]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Pythagorean identities:"})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"\u03B8"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msup,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"sec"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"\u03B8"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\sin^2\\theta + \\cos^2\\theta = 1 \\qquad \\tan^2\\theta + 1 = \\sec^2\\theta"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.9552em",verticalAlign:"-0.0833em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8719em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.1208em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8141em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.9024em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop",children:"tan"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8191em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.068em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8141em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop",children:"sec"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0278em"},children:"\u03B8"})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Compound angle formulas:"})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\xB1"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{children:"\xB1"}),(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"B"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\sin(A \\pm B) = \\sin A\\cos B \\pm \\cos A\\sin B"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\xB1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7667em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\xB1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\xB1"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{children:"\u2213"}),(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"B"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\cos(A \\pm B) = \\cos A\\cos B \\mp \\sin A\\sin B"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\xB1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7667em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2213"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\xB1"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\xB1"}),(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"B"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{children:"\u2213"}),(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"B"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\tan(A \\pm B) = \\frac{\\tan A \\pm \\tan B}{1 \\mp \\tan A\\tan B}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mop",children:"tan"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\xB1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2757em",verticalAlign:"-0.4033em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8723em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mtight",children:"1"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2213"}),(0,e.jsxs)(s.span,{className:"mop mtight",children:[(0,e.jsx)(s.span,{className:"mtight",children:"t"}),(0,e.jsx)(s.span,{className:"mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mtight",children:"n"})]}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"A"}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsxs)(s.span,{className:"mop mtight",children:[(0,e.jsx)(s.span,{className:"mtight",children:"t"}),(0,e.jsx)(s.span,{className:"mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mtight",children:"n"})]}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0502em"},children:"B"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsxs)(s.span,{className:"mop mtight",children:[(0,e.jsx)(s.span,{className:"mtight",children:"t"}),(0,e.jsx)(s.span,{className:"mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mtight",children:"n"})]}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"A"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\xB1"}),(0,e.jsxs)(s.span,{className:"mop mtight",children:[(0,e.jsx)(s.span,{className:"mtight",children:"t"}),(0,e.jsx)(s.span,{className:"mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mtight",children:"n"})]}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0502em"},children:"B"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.4033em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Double angle formulas:"})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"A"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\sin 2A = 2\\sin A\\cos A"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msup,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\cos 2A = \\cos^2 A - \\sin^2 A = 1 - 2\\sin^2 A = 2\\cos^2 A - 1"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8719em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8719em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.1208em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7278em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8719em"}}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8719em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.1208em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mi,{children:"A"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"tan"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"A"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\tan 2A = \\frac{2\\tan A}{1 - \\tan^2 A}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mop",children:"tan"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2757em",verticalAlign:"-0.4033em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8723em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mtight",children:"1"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsxs)(s.span,{className:"mop mtight",children:[(0,e.jsxs)(s.span,{className:"mop mtight",children:[(0,e.jsx)(s.span,{className:"mtight",children:"t"}),(0,e.jsx)(s.span,{className:"mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mtight",children:"n"})]}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7463em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.786em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"A"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mtight",children:"2"}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsxs)(s.span,{className:"mop mtight",children:[(0,e.jsx)(s.span,{className:"mtight",children:"t"}),(0,e.jsx)(s.span,{className:"mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mtight",children:"n"})]}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"A"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.4033em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsx)(s.h2,{id:"section-3-introductory-calculus",children:"Section 3: Introductory Calculus"}),` `,(0,e.jsx)(s.h3,{id:"a-differentiation",children:"A. Differentiation"}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f(x)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})}),(0,e.jsx)(s.th,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u2032"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f'(x)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0019em",verticalAlign:"-0.25em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7519em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u2032"})})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mi,{children:"n"})]})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"x^n"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6644em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"1"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"nx^{n-1}"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8141em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"n"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})]})})]})})})})})]})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\sin(ax)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a\\cos(ax)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\cos(ax)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"-a\\sin(ax)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})})]})]})]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Chain rule"})," (let ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"u"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"g"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"u = g(x)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"u"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"g"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]}),", ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"u"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"y = f(u)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"u"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]}),"):"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"y"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"y"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"u"})]})]}),(0,e.jsx)(s.mo,{children:"\u22C5"}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"u"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\frac{dy}{dx} = \\frac{dy}{du} \\cdot \\frac{du}{dx}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2772em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9322em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.4461em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"y"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2772em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9322em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"u"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.4461em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"y"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u22C5"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2251em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8801em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"u"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Product rule"})," (",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"u"}),(0,e.jsx)(s.mi,{children:"v"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"y = uv"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"uv"})]})]})]}),"):"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"y"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"u"}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"v"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"v"}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"u"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\frac{dy}{dx} = u\\frac{dv}{dx} + v\\frac{du}{dx}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2772em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9322em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.4461em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"y"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2251em",verticalAlign:"-0.345em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"u"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8801em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"v"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2251em",verticalAlign:"-0.345em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8801em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"u"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Quotient rule"})," (",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mi,{children:"u"}),(0,e.jsx)(s.mi,{children:"v"})]})})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"y = \\dfrac{u}{v}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.7936em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.1076em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord mathnormal",children:"u"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]}),"):"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"y"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"v"}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"u"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]})]})}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"u"}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"v"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]})]})})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"v"}),(0,e.jsx)(s.mn,{children:"2"})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\frac{dy}{dx} = \\frac{v\\dfrac{du}{dx} - u\\dfrac{dv}{dx}}{v^2}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2772em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9322em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.4461em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"y"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.7124em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"2.3674em"},children:[(0,e.jsxs)(s.span,{style:{top:"-3.0264em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.3714em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"v"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7463em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.786em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.6014em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.3714em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-4.3674em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.3714em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"v"}),(0,e.jsxs)(s.span,{className:"mord sizing reset-size3 size6",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3714em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"u"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"u"}),(0,e.jsxs)(s.span,{className:"mord sizing reset-size3 size6",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3714em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Stationary points:"})," set ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u2032"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f'(x) = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0019em",verticalAlign:"-0.25em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7519em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u2032"})})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]}),", solve for ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"x"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"x"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})})]}),", find ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"y"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"y"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"})]})})]}),"."]}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:"Second derivative at stationary point"}),(0,e.jsx)(s.th,{children:"Nature"})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{mathvariant:"normal",children:"\u2032"}),(0,e.jsx)(s.mo,{mathvariant:"normal",children:"\u2032"})]})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:">"}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f''(x) > 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0019em",verticalAlign:"-0.25em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7519em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u2032\u2032"})})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:">"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]})}),(0,e.jsx)(s.td,{children:"Minimum"})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{mathvariant:"normal",children:"\u2032"}),(0,e.jsx)(s.mo,{mathvariant:"normal",children:"\u2032"})]})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"<"}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f''(x) < 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0019em",verticalAlign:"-0.25em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7519em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u2032\u2032"})})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"<"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]})}),(0,e.jsx)(s.td,{children:"Maximum"})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{mathvariant:"normal",children:"\u2032"}),(0,e.jsx)(s.mo,{mathvariant:"normal",children:"\u2032"})]})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f''(x) = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0019em",verticalAlign:"-0.25em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7519em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u2032\u2032"})})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]})}),(0,e.jsxs)(s.td,{children:["Inconclusive; use sign-change test on ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u2032"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f'(x)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0019em",verticalAlign:"-0.25em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7519em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u2032"})})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]})]})]})]})]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Tangent and normal"})," at ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"(x_0, y_0)"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"0"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"-0.0359em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"0"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})})]}),": tangent gradient ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mi,{children:"T"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u2032"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"m_T = f'(x_0)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"m"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3283em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.1389em"},children:"T"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.0019em",verticalAlign:"-0.25em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7519em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u2032"})})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"0"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]}),"; normal gradient ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mi,{children:"N"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"m"}),(0,e.jsx)(s.mi,{children:"T"})]})]})})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"m_N = -\\dfrac{1}{m_T}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"m"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3283em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.109em"},children:"N"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.1574em",verticalAlign:"-0.836em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u2212"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3214em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"m"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3283em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.1389em"},children:"T"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mord",children:"1"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.836em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]}),"."]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Connected rates:"})," ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"y"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"t"})]})]})}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"y"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]})]})}),(0,e.jsx)(s.mo,{children:"\u22C5"}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"t"})]})]})})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\dfrac{dy}{dt} = \\dfrac{dy}{dx} \\cdot \\dfrac{dx}{dt}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.0574em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3714em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"t"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.0574em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3714em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u22C5"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.0574em",verticalAlign:"-0.686em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3714em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"t"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.686em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})]}),` `,(0,e.jsx)(s.h3,{id:"b-integration",children:"B. Integration"}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u222B"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mn,{children:"1"})]})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mn,{children:"1"})]})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"c"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mo,{mathvariant:"normal",children:"\u2260"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\int x^n\\,dx = \\frac{x^{n+1}}{n+1} + c \\qquad (n \\neq -1)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1111em",verticalAlign:"-0.3061em"}}),(0,e.jsx)(s.span,{className:"mop op-symbol small-op",style:{marginRight:"0.1945em",position:"relative",top:"-0.0006em"},children:"\u222B"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.4213em",verticalAlign:"-0.4033em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.0179em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"+"}),(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8913em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.931em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"+"}),(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})]})})]})})})})})]})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.4033em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"n"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsxs)(s.span,{className:"mrel",children:[(0,e.jsx)(s.span,{className:"mrel",children:(0,e.jsx)(s.span,{className:"mord vbox",children:(0,e.jsx)(s.span,{className:"thinbox",children:(0,e.jsxs)(s.span,{className:"rlap",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"inner",children:(0,e.jsx)(s.span,{className:"mord",children:(0,e.jsx)(s.span,{className:"mrel",children:"\uE020"})})}),(0,e.jsx)(s.span,{className:"fix"})]})})})}),(0,e.jsx)(s.span,{className:"mspace nobreak"}),(0,e.jsx)(s.span,{className:"mrel",children:"="})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord",children:"1"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u222B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mn,{children:"1"})]})]})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"n"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"c"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\int (ax + b)^n\\,dx = \\frac{(ax + b)^{n+1}}{a(n+1)} + c"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1111em",verticalAlign:"-0.3061em"}}),(0,e.jsx)(s.span,{className:"mop op-symbol small-op",style:{marginRight:"0.1945em",position:"relative",top:"-0.0006em"},children:"\u222B"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsxs)(s.span,{className:"mclose",children:[(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.6644em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.6289em",verticalAlign:"-0.52em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.1089em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mopen mtight",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"+"}),(0,e.jsx)(s.span,{className:"mord mtight",children:"1"}),(0,e.jsx)(s.span,{className:"mclose mtight",children:")"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.485em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mopen mtight",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"+"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"b"}),(0,e.jsxs)(s.span,{className:"mclose mtight",children:[(0,e.jsx)(s.span,{className:"mclose mtight",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8913em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.931em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"+"}),(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})]})})]})})})})})]})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.52em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u222B"}),(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"c"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mo,{children:"\u222B"}),(0,e.jsx)(s.mi,{children:"cos"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.mi,{children:"sin"}),(0,e.jsx)(s.mo,{children:"\u2061"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"c"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\int \\sin(ax)\\,dx = -\\frac{1}{a}\\cos(ax) + c \\qquad \\int \\cos(ax)\\,dx = \\frac{1}{a}\\sin(ax) + c"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1111em",verticalAlign:"-0.3061em"}}),(0,e.jsx)(s.span,{className:"mop op-symbol small-op",style:{marginRight:"0.1945em",position:"relative",top:"-0.0006em"},children:"\u222B"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1901em",verticalAlign:"-0.345em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u2212"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8451em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1111em",verticalAlign:"-0.3061em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop op-symbol small-op",style:{marginRight:"0.1945em",position:"relative",top:"-0.0006em"},children:"\u222B"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"cos"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1901em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8451em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mop",children:"sin"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"c"})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Area under a curve"})," between ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"a"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"x = a"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})]})]})]})," and ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"b"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"x = b"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6944em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"})]})]})]}),":"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msubsup,{children:[(0,e.jsx)(s.mo,{children:"\u222B"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"b"})]}),(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"A = \\int_a^b y\\,dx"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.3998em",verticalAlign:"-0.3558em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop op-symbol small-op",style:{marginRight:"0.1945em",position:"relative",top:"-0.0006em"},children:"\u222B"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.044em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.3442em",marginLeft:"-0.1945em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.2579em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"b"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3558em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Area between two curves"})," (",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"\u2265"}),(0,e.jsx)(s.mi,{children:"g"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"f(x) \\geq g(x)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"\u2265"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"g"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})," on ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:"["}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{separator:"true",children:","}),(0,e.jsx)(s.mi,{children:"b"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"]"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"[a,b]"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mopen",children:"["}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mpunct",children:","}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"b"}),(0,e.jsx)(s.span,{className:"mclose",children:"]"})]})})]}),"):"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msubsup,{children:[(0,e.jsx)(s.mo,{children:"\u222B"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"b"})]}),(0,e.jsx)(s.mo,{fence:"true",stretchy:"true",minsize:"1.2em",maxsize:"1.2em",children:"["}),(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"g"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{fence:"true",stretchy:"true",minsize:"1.2em",maxsize:"1.2em",children:"]"}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"A = \\int_a^b \\bigl[f(x) - g(x)\\bigr]\\,dx"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.3998em",verticalAlign:"-0.3558em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop op-symbol small-op",style:{marginRight:"0.1945em",position:"relative",top:"-0.0006em"},children:"\u222B"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.044em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.3442em",marginLeft:"-0.1945em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.2579em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"b"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3558em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mopen",children:(0,e.jsx)(s.span,{className:"delimsizing size1",children:"["})}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2em",verticalAlign:"-0.35em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"g"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mclose",children:(0,e.jsx)(s.span,{className:"delimsizing size1",children:"]"})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Volume of revolution"})," about the ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"x"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"x"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})})]}),"-axis:"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"V"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"\u03C0"}),(0,e.jsxs)(s.msubsup,{children:[(0,e.jsx)(s.mo,{children:"\u222B"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"b"})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"y"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"x"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"V = \\pi\\int_a^b y^2\\,dx"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.2222em"},children:"V"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.3998em",verticalAlign:"-0.3558em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"\u03C0"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mop",children:[(0,e.jsx)(s.span,{className:"mop op-symbol small-op",style:{marginRight:"0.1945em",position:"relative",top:"-0.0006em"},children:"\u222B"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.044em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.3442em",marginLeft:"-0.1945em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"a"})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.2579em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"b"})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3558em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"y"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]})]})]})}),` `,(0,e.jsx)(s.h3,{id:"kinematics",children:"Kinematics"}),` `,(0,e.jsxs)(s.p,{children:["Displacement ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"s"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"s"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"s"})]})})]}),", velocity ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"v"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"v"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"})]})})]}),", acceleration ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"a"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"a"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"})]})})]}),", time ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"t"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"t"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6151em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"t"})]})})]}),":"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"v"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"s"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"t"})]})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"v"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"t"})]})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mi,{children:"s"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"t"}),(0,e.jsx)(s.mn,{children:"2"})]})]})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"v = \\frac{ds}{dt} \\qquad a = \\frac{dv}{dt} = \\frac{d^2s}{dt^2}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2251em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8801em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"t"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"s"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2251em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8801em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"t"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.0359em"},children:"v"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.3629em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.0179em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"t"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7463em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.786em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"d"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8913em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.931em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"s"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"s"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mo,{children:"\u222B"}),(0,e.jsx)(s.mi,{children:"v"}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"t"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"v"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mo,{children:"\u222B"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mtext,{children:"\u2009"}),(0,e.jsx)(s.mi,{children:"d"}),(0,e.jsx)(s.mi,{children:"t"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"s = \\int v\\,dt \\qquad v = \\int a\\,dt"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"s"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1111em",verticalAlign:"-0.3061em"}}),(0,e.jsx)(s.span,{className:"mop op-symbol small-op",style:{marginRight:"0.1945em",position:"relative",top:"-0.0006em"},children:"\u222B"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"t"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1111em",verticalAlign:"-0.3061em"}}),(0,e.jsx)(s.span,{className:"mop op-symbol small-op",style:{marginRight:"0.1945em",position:"relative",top:"-0.0006em"},children:"\u222B"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"d"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"t"})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"SUVAT equations"})," (uniform acceleration only; ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"u"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"u"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"u"})]})})]})," = initial velocity, ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"s"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"s"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"s"})]})})]})," = displacement):"]}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"v"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"u"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"t"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"s"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"u"}),(0,e.jsx)(s.mi,{children:"t"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mn,{children:"2"})]})}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"t"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"v"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"u"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mn,{children:"2"}),(0,e.jsx)(s.mi,{children:"a"}),(0,e.jsx)(s.mi,{children:"s"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"s"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mn,{children:"2"})]})}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"u"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"v"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mi,{children:"t"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"v = u + at \\qquad s = ut + \\tfrac{1}{2}at^2 \\qquad v^2 = u^2 + 2as \\qquad s = \\tfrac{1}{2}(u + v)t"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"u"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6151em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"t"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"s"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6984em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"u"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"t"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1901em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8451em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"t"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"u"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"2"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"a"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"s"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"s"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.1901em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8451em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"u"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"t"})]})]})]})}),` `,(0,e.jsxs)(s.p,{children:["Particle is stationary when ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"v"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"0"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"v = 0"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"})]})]})]}),". Direction changes when ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"v"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"v"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.4306em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0359em"},children:"v"})]})})]})," changes sign."]}),` `,(0,e.jsx)(s.h2,{id:"section-4-probability-and-statistics",children:"Section 4: Probability and Statistics"}),` `,(0,e.jsx)(s.h3,{id:"a-data-representation-and-analysis",children:"A. Data Representation and Analysis"}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Mean:"})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.mover,{accent:"true",children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u02C9"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u2211"}),(0,e.jsx)(s.mi,{children:"x"})]}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.mspace,{width:"1em"}),(0,e.jsx)(s.mtext,{children:"(ungrouped)"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsxs)(s.mover,{accent:"true",children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u02C9"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u2211"}),(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mi,{children:"x"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u2211"}),(0,e.jsx)(s.mi,{children:"f"})]})]}),(0,e.jsx)(s.mspace,{width:"1em"}),(0,e.jsx)(s.mtext,{children:"(grouped)"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\bar{x} = \\frac{\\sum x}{n} \\quad \\text{(ungrouped)} \\qquad \\bar{x} = \\frac{\\sum fx}{\\sum f} \\quad \\text{(grouped)}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.5678em"}}),(0,e.jsx)(s.span,{className:"mord accent",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.5678em"},children:[(0,e.jsxs)(s.span,{style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]}),(0,e.jsxs)(s.span,{style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"accent-body",style:{left:"-0.2222em"},children:(0,e.jsx)(s.span,{className:"mord",children:"\u02C9"})})]})]})})})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.355em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.01em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.485em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mop op-symbol small-op mtight",style:{position:"relative",top:"0em"},children:"\u2211"}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"1em"}}),(0,e.jsx)(s.span,{className:"mord text",children:(0,e.jsx)(s.span,{className:"mord",children:"(ungrouped)"})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord accent",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.5678em"},children:[(0,e.jsxs)(s.span,{style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"})]}),(0,e.jsxs)(s.span,{style:{top:"-3em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"accent-body",style:{left:"-0.2222em"},children:(0,e.jsx)(s.span,{className:"mord",children:"\u02C9"})})]})]})})})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.53em",verticalAlign:"-0.52em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.01em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mop op-symbol small-op mtight",style:{position:"relative",top:"0em"},children:"\u2211"}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.1076em"},children:"f"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.485em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mop op-symbol small-op mtight",style:{position:"relative",top:"0em"},children:"\u2211"}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.52em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"1em"}}),(0,e.jsx)(s.span,{className:"mord text",children:(0,e.jsx)(s.span,{className:"mord",children:"(grouped)"})})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Variance and standard deviation:"})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"S"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u2211"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.mover,{accent:"true",children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u02C9"})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsx)(s.mi,{children:"n"})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"S"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.msqrt,{children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u2211"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.mover,{accent:"true",children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u02C9"})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsx)(s.mi,{children:"n"})]})}),(0,e.jsx)(s.mspace,{width:"1em"}),(0,e.jsx)(s.mtext,{children:"(ungrouped)"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"S^2 = \\frac{\\sum(x - \\bar{x})^2}{n} \\qquad S = \\sqrt{\\frac{\\sum(x - \\bar{x})^2}{n}} \\quad \\text{(ungrouped)}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8141em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0576em"},children:"S"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.4539em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.1089em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.485em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mop op-symbol small-op mtight",style:{position:"relative",top:"0em"},children:"\u2211"}),(0,e.jsx)(s.span,{className:"mopen mtight",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord accent mtight",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.5678em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.7em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]}),(0,e.jsxs)(s.span,{style:{top:"-2.7em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"accent-body",style:{left:"-0.2222em"},children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u02C9"})})]})]})})})}),(0,e.jsxs)(s.span,{className:"mclose mtight",children:[(0,e.jsx)(s.span,{className:"mclose mtight",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8913em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.931em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0576em"},children:"S"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.84em",verticalAlign:"-0.5225em"}}),(0,e.jsx)(s.span,{className:"mord sqrt",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.3175em"},children:[(0,e.jsxs)(s.span,{className:"svg-align",style:{top:"-3.8em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.8em"}}),(0,e.jsx)(s.span,{className:"mord",style:{paddingLeft:"1em"},children:(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.01em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"n"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.485em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mop op-symbol small-op mtight",style:{position:"relative",top:"0em"},children:"\u2211"}),(0,e.jsx)(s.span,{className:"mopen mtight",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord accent mtight",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.5678em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.7em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]}),(0,e.jsxs)(s.span,{style:{top:"-2.7em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"accent-body",style:{left:"-0.2222em"},children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u02C9"})})]})]})})})}),(0,e.jsxs)(s.span,{className:"mclose mtight",children:[(0,e.jsx)(s.span,{className:"mclose mtight",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7463em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.786em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.2775em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3.8em"}}),(0,e.jsx)(s.span,{className:"hide-tail",style:{minWidth:"1.02em",height:"1.88em"},children:(0,e.jsx)(s.svg,{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice",children:(0,e.jsx)(s.path,{d:`M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z`})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.5225em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"1em"}}),(0,e.jsx)(s.span,{className:"mord text",children:(0,e.jsx)(s.span,{className:"mord",children:"(ungrouped)"})})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"S"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u2211"}),(0,e.jsx)(s.mi,{children:"f"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.mover,{accent:"true",children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mo,{children:"\u02C9"})]}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mo,{children:"\u2211"}),(0,e.jsx)(s.mi,{children:"f"})]})]}),(0,e.jsx)(s.mspace,{width:"1em"}),(0,e.jsx)(s.mtext,{children:"(grouped)"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"S^2 = \\frac{\\sum f(x - \\bar{x})^2}{\\sum f} \\quad \\text{(grouped)}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8141em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0576em"},children:"S"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8141em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.6289em",verticalAlign:"-0.52em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.1089em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mop op-symbol small-op mtight",style:{position:"relative",top:"0em"},children:"\u2211"}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.1076em"},children:"f"})]})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.485em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mop op-symbol small-op mtight",style:{position:"relative",top:"0em"},children:"\u2211"}),(0,e.jsx)(s.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.1076em"},children:"f"}),(0,e.jsx)(s.span,{className:"mopen mtight",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mord accent mtight",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.5678em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.7em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"x"})]}),(0,e.jsxs)(s.span,{style:{top:"-2.7em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"accent-body",style:{left:"-0.2222em"},children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u02C9"})})]})]})})})}),(0,e.jsxs)(s.span,{className:"mclose mtight",children:[(0,e.jsx)(s.span,{className:"mclose mtight",children:")"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.8913em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.931em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.52em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"1em"}}),(0,e.jsx)(s.span,{className:"mord text",children:(0,e.jsx)(s.span,{className:"mord",children:"(grouped)"})})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Measures of spread:"})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mtext,{children:"Range"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mi,{children:"max"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"x"}),(0,e.jsx)(s.mi,{children:"min"}),(0,e.jsx)(s.mo,{children:"\u2061"})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mtext,{children:"IQR"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"3"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mtext,{children:"Semi-IQR"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"3"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"1"})]})]}),(0,e.jsx)(s.mn,{children:"2"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"\\text{Range} = x_{\\max} - x_{\\min} \\qquad \\text{IQR} = Q_3 - Q_1 \\qquad \\text{Semi-IQR} = \\frac{Q_3 - Q_1}{2}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsx)(s.span,{className:"mord text",children:(0,e.jsx)(s.span,{className:"mord",children:"Range"})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7333em",verticalAlign:"-0.15em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.1514em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsxs)(s.span,{className:"mop mtight",children:[(0,e.jsx)(s.span,{className:"mtight",children:"m"}),(0,e.jsx)(s.span,{className:"mtight",children:"a"}),(0,e.jsx)(s.span,{className:"mtight",children:"x"})]})})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"x"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3175em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsxs)(s.span,{className:"mop mtight",children:[(0,e.jsx)(s.span,{className:"mtight",children:"m"}),(0,e.jsx)(s.span,{className:"mtight",children:"i"}),(0,e.jsx)(s.span,{className:"mtight",children:"n"})]})})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord text",children:(0,e.jsx)(s.span,{className:"mord",children:"IQR"})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"3"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord text",children:(0,e.jsx)(s.span,{className:"mord",children:"Semi-IQR"})}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2694em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.9244em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.4461em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"0em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"3"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mbin mtight",children:"\u2212"}),(0,e.jsxs)(s.span,{className:"mord mtight",children:[(0,e.jsx)(s.span,{className:"mord mathnormal mtight",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3173em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.357em",marginLeft:"0em",marginRight:"0.0714em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.5em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size3 size1 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.143em"},children:(0,e.jsx)(s.span,{})})})]})})]})]})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})}),` `,(0,e.jsx)(s.p,{children:(0,e.jsx)(s.strong,{children:"Skewness:"})}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:"Box plot pattern"}),(0,e.jsx)(s.th,{children:"Skew"}),(0,e.jsx)(s.th,{children:"Order of averages"})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsxs)(s.td,{children:["Right whisker longer / ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"3"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:">"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"1"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"Q_3 - Q_2 > Q_2 - Q_1"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"3"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:">"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]})]})]})]})]}),(0,e.jsx)(s.td,{children:"Positive"}),(0,e.jsx)(s.td,{children:"Mode < Median < Mean"})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsxs)(s.td,{children:["Left whisker longer / ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"3"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"<"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"2"})]}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsxs)(s.msub,{children:[(0,e.jsx)(s.mi,{children:"Q"}),(0,e.jsx)(s.mn,{children:"1"})]})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"Q_3 - Q_2 < Q_2 - Q_1"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"3"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"<"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"2"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"Q"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.3011em"},children:(0,e.jsxs)(s.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"1"})})]})}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.15em"},children:(0,e.jsx)(s.span,{})})})]})})]})]})]})]})]}),(0,e.jsx)(s.td,{children:"Negative"}),(0,e.jsx)(s.td,{children:"Mean < Median < Mode"})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:"Symmetric"}),(0,e.jsx)(s.td,{children:"None"}),(0,e.jsx)(s.td,{children:"Mode = Median = Mean"})]})]})]}),` `,(0,e.jsx)(s.h3,{id:"b-probability-theory",children:"B. Probability Theory"}),` `,(0,e.jsx)(s.p,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsxs)(s.mfrac,{children:[(0,e.jsx)(s.mtext,{children:"favourable\xA0outcomes"}),(0,e.jsx)(s.mtext,{children:"total\xA0outcomes"})]}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsxs)(s.msup,{children:[(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u2032"})]}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"1"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mspace,{width:"2em"}),(0,e.jsx)(s.mn,{children:"0"}),(0,e.jsx)(s.mo,{children:"\u2264"}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"\u2264"}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"P(A) = \\frac{\\text{favourable outcomes}}{\\text{total outcomes}} \\qquad P(A') = 1 - P(A) \\qquad 0 \\leq P(A) \\leq 1"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1.2251em",verticalAlign:"-0.345em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"0.8801em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.655em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord text mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"total\xA0outcomes"})})})})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.394em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord text mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"favourable\xA0outcomes"})})})})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.345em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"msupsub",children:(0,e.jsx)(s.span,{className:"vlist-t",children:(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.7519em"},children:(0,e.jsxs)(s.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,e.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:(0,e.jsx)(s.span,{className:"mord mtight",children:"\u2032"})})})]})})})})})]}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.7278em",verticalAlign:"-0.0833em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"2em"}}),(0,e.jsx)(s.span,{className:"mord",children:"0"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"\u2264"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"\u2264"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})]})]})}),` `,(0,e.jsxs)(s.table,{children:[(0,e.jsx)(s.thead,{children:(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.th,{children:"Rule"}),(0,e.jsx)(s.th,{children:"Formula"})]})}),(0,e.jsxs)(s.tbody,{children:[(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:"Addition (general)"}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\u222A"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"\u2212"}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\u2229"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"P(A \\cup B) = P(A) + P(B) - P(A \\cap B)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u222A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2212"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2229"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsxs)(s.td,{children:["Mutually exclusive (",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\u2229"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{mathvariant:"normal",children:"\u2205"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"A \\cap B = \\emptyset"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2229"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.8056em",verticalAlign:"-0.0556em"}}),(0,e.jsx)(s.span,{className:"mord",children:"\u2205"})]})]})]}),")"]}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\u222A"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"+"}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"P(A \\cup B) = P(A) + P(B)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u222A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"+"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:"Conditional"}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\u2223"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,e.jsxs)(s.mfrac,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\u2229"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]})]})})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"P(A \\mid B) = \\dfrac{P(A \\cap B)}{P(B)}"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"\u2223"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"2.363em",verticalAlign:"-0.936em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mopen nulldelimiter"}),(0,e.jsx)(s.span,{className:"mfrac",children:(0,e.jsxs)(s.span,{className:"vlist-t vlist-t2",children:[(0,e.jsxs)(s.span,{className:"vlist-r",children:[(0,e.jsxs)(s.span,{className:"vlist",style:{height:"1.427em"},children:[(0,e.jsxs)(s.span,{style:{top:"-2.314em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]}),(0,e.jsxs)(s.span,{style:{top:"-3.23em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsx)(s.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,e.jsxs)(s.span,{style:{top:"-3.677em"},children:[(0,e.jsx)(s.span,{className:"pstrut",style:{height:"3em"}}),(0,e.jsxs)(s.span,{className:"mord",children:[(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2229"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]}),(0,e.jsx)(s.span,{className:"vlist-s",children:"\u200B"})]}),(0,e.jsx)(s.span,{className:"vlist-r",children:(0,e.jsx)(s.span,{className:"vlist",style:{height:"0.936em"},children:(0,e.jsx)(s.span,{})})})]})}),(0,e.jsx)(s.span,{className:"mclose nulldelimiter"})]})]})]})]})})]}),(0,e.jsxs)(s.tr,{children:[(0,e.jsx)(s.td,{children:"Independent"}),(0,e.jsx)(s.td,{children:(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\u2229"}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"\u22C5"}),(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"B"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"P(A \\cap B) = P(A) \\cdot P(B)"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2229"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u22C5"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"}),(0,e.jsx)(s.span,{className:"mclose",children:")"})]})]})]})})]})]})]}),` `,(0,e.jsx)(s.p,{children:"Two non-zero-probability events cannot be simultaneously mutually exclusive and independent."}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Tree diagrams:"})," multiply along branches for intersection probabilities; add across branches for union. The syllabus restricts trees to two initial branches."]}),` `,(0,e.jsxs)(s.p,{children:[(0,e.jsx)(s.strong,{children:"Venn diagrams:"})," restricted to two sets; the four regions (",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"A"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"A"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"})]})})]})," only, ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsx)(s.mrow,{children:(0,e.jsx)(s.mi,{children:"B"})}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"B"})]})})}),(0,e.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"})]})})]})," only, ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"A"}),(0,e.jsx)(s.mo,{children:"\u2229"}),(0,e.jsx)(s.mi,{children:"B"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"A \\cap B"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",children:"A"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,e.jsx)(s.span,{className:"mbin",children:"\u2229"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0502em"},children:"B"})]})]})]}),", neither) must sum to ",(0,e.jsxs)(s.span,{className:"katex",children:[(0,e.jsx)(s.span,{className:"katex-mathml",children:(0,e.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,e.jsxs)(s.semantics,{children:[(0,e.jsxs)(s.mrow,{children:[(0,e.jsx)(s.mi,{children:"P"}),(0,e.jsx)(s.mo,{stretchy:"false",children:"("}),(0,e.jsx)(s.mi,{children:"S"}),(0,e.jsx)(s.mo,{stretchy:"false",children:")"}),(0,e.jsx)(s.mo,{children:"="}),(0,e.jsx)(s.mn,{children:"1"})]}),(0,e.jsx)(s.annotation,{encoding:"application/x-tex",children:"P(S) = 1"})]})})}),(0,e.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.1389em"},children:"P"}),(0,e.jsx)(s.span,{className:"mopen",children:"("}),(0,e.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.0576em"},children:"S"}),(0,e.jsx)(s.span,{className:"mclose",children:")"}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,e.jsx)(s.span,{className:"mrel",children:"="}),(0,e.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,e.jsxs)(s.span,{className:"base",children:[(0,e.jsx)(s.span,{className:"strut",style:{height:"0.6444em"}}),(0,e.jsx)(s.span,{className:"mord",children:"1"})]})]})]}),"."]})]})}function d(a={}){let{wrapper:s}=a.components||{};return s?(0,e.jsx)(s,{...a,children:(0,e.jsx)(h,{...a})}):h(a)}return w(R);})(); ;return Component;

French

French