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tests0002.html
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Braille and Math Speech Tests 0002 (Mostly from Differential Calculus)</title>
<style>
/*Custom Fonts*/
@font-face {
font-family: 'Sim Braille';
src: url('fonts/simbraille-webfont.woff2') format('woff2'),
url('fonts/simbbraille-webfont.woff') format('woff');
font-weight: normal;
font-style: normal;
}
html, body {
background-color: #1e1e1e;
color: #d4d4d4;
}
body {
margin: 0;
}
fieldset {
margin: 0.5rem;
padding: 0.5rem;
}
div.exampleContainer {
border: thin dashed gray;
margin: 0.5rem;
padding: 0.5rem;
}
h3.example {
margin: 0;
}
div.exampleContainer p {
font-family: 'Consolas';
}
div.resultContainer {
display: flex;
flex-wrap: wrap;
}
div.outerRenderedMathContainer {
flex-grow: 1;
width: 33%;
margin-right: 0.5rem;
min-width: 300px;
}
div.outerMathSrcContainer {
flex-grow: 1;
width: 65%;
}
@media only screen and (max-width: 768px) {
div.resultContainer {
display: block;
}
div.outerRenderedMathContainer {
display: block;
width: 95%;
}
div.outerMathSrcContainer {
display: block;
width: 95%;
margin-top: 1.3rem;
}
}
div.resultContainer h4 {
margin: 0.5rem 0 0.1rem 0;
}
div.renderedMathContainer {
padding: 5px;
border-radius: 5px;
background-color: #d4d4d4;
color: black;
width: 95%;
}
div.mathSrcContainer pre {
margin: 0;
}
div.speechTextAndBrailleContainer p {
margin-top: 0.1rem;
margin-left: 0.5rem;
}
div.speechTextAndBrailleContainer > p.mathCatBraille.braille, div.speechTextAndBrailleContainer > p.sreBraille.braille {
overflow-wrap: break-word;
font-family: 'Sim Braille', Consolas;
font-size: 125%;
}
span.deletion {
color: red;
}
span.insertion {
border-left: thin solid green;
}
.tagName { color: #68AEC5; }
.tagChar { color: #969696}
.tagValueQuote { color: #d08c0d; }
.tagEqSign { color: #c49e57; }
.tagAttrName { color: #68CDFE; }
.tagAttrValue { color: rgb(255, 115, 0)}
.tagText { color: #D4D4D4}
</style>
<script src="patienceDiff.js"></script>
<script>
(()=>{
math=[{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>ln</mi><mo>⁡</mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></math>","mathSpeak":"StartFraction d Over d x EndFraction ln x equals StartFraction 1 Over x EndFraction","sreBraille":"⠹⠙⠌⠙⠭⠼⠇⠝⠀⠭⠀⠨⠅⠀⠹⠂⠌⠭⠼","clearSpeak":"d over d x l n x, equals 1 over x","mathCatBraille":"⠹⠙⠌⠙⠭⠼⠇⠝⠀⠭⠀⠨⠅⠀⠹⠂⠌⠭⠼"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi><mo>=</mo><mi>sin</mi><mo>⁡</mo><mfrac><mi>a</mi><mi>b</mi></mfrac></math>","mathSpeak":"x equals sine StartFraction a Over b EndFraction","sreBraille":"⠭⠀⠨⠅⠀⠎⠊⠝⠀⠹⠁⠌⠃⠼","clearSpeak":"x equals sine of eigh over b","mathCatBraille":"⠭⠀⠨⠅⠀⠎⠊⠝⠀⠹⠁⠌⠃⠼"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>sin</mi><mo>⁡</mo><mi>a</mi></mrow><mi>b</mi></mfrac></math>","mathSpeak":"x equals StartFraction sine a Over b EndFraction","sreBraille":"⠭⠀⠨⠅⠀⠹⠎⠊⠝⠀⠁⠌⠃⠼","clearSpeak":"x equals sine of eigh over b","mathCatBraille":"⠭⠀⠨⠅⠀⠹⠎⠊⠝⠀⠁⠌⠃⠼"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi><mo>=</mo><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mi>a</mi><mo>−</mo><mi>b</mi><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mo>−</mo><mi>c</mi></math>","mathSpeak":"x equals StartAbsoluteValue a minus b EndAbsoluteValue minus c","sreBraille":"⠭⠀⠨⠅⠀⠳⠁⠤⠃⠳⠤⠉","clearSpeak":"x equals, the absolute value of eigh minus b, minus c","mathCatBraille":"⠭⠀⠨⠅⠀⠳⠁⠤⠃⠳⠤⠉"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi><mo>=</mo><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mi>a</mi><mo>−</mo><mi>b</mi><mo>−</mo><mi>c</mi><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow></math>","mathSpeak":"x equals StartAbsoluteValue a minus b minus c EndAbsoluteValue","sreBraille":"⠭⠀⠨⠅⠀⠳⠁⠤⠃⠤⠉⠳","clearSpeak":"x equals, the absolute value of eigh minus b minus c","mathCatBraille":"⠭⠀⠨⠅⠀⠳⠁⠤⠃⠤⠉⠳"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>+</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></msqrt></math>","mathSpeak":"f left parenthesis x right parenthesis equals StartFraction 1 Over x squared EndFraction plus StartRoot x squared minus 1 EndRoot","sreBraille":"⠋⠷⠭⠾⠀⠨⠅⠀⠹⠂⠌⠭⠘⠆⠐⠼⠬⠜⠭⠘⠆⠐⠤⠂⠻","clearSpeak":"f of x equals; the fraction with numerator 1; and denominator x squared; plus, the square root of x squared minus 1;","mathCatBraille":"⠋⠷⠭⠾⠀⠨⠅⠀⠹⠂⠌⠭⠘⠆⠐⠼⠬⠜⠭⠘⠆⠐⠤⠂⠻"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></msqrt></mrow></mfrac></math>","mathSpeak":"f left parenthesis x right parenthesis equals StartFraction 1 Over x squared plus StartRoot x squared minus 1 EndRoot EndFraction","sreBraille":"⠋⠷⠭⠾⠀⠨⠅⠀⠹⠂⠌⠭⠘⠆⠐⠬⠜⠭⠘⠆⠐⠤⠂⠻⠼","clearSpeak":"f of x equals; the fraction with numerator 1; and denominator x squared plus, the square root of x squared minus 1;","mathCatBraille":"⠋⠷⠭⠾⠀⠨⠅⠀⠹⠂⠌⠭⠘⠆⠐⠬⠜⠭⠘⠆⠐⠤⠂⠻⠼"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi>cos</mi><mo>⁡</mo><mo stretchy=\"false\">(</mo><mi>α</mi><mo>±</mo><mi>β</mi><mo stretchy=\"false\">)</mo></mtd><mtd><mi></mi><mo>=</mo><mi>cos</mi><mo>⁡</mo><mo stretchy=\"false\">(</mo><mi>α</mi><mo stretchy=\"false\">)</mo><mi>cos</mi><mo>⁡</mo><mo stretchy=\"false\">(</mo><mi>β</mi><mo stretchy=\"false\">)</mo><mo>∓</mo><mi>sin</mi><mo>⁡</mo><mo stretchy=\"false\">(</mo><mi>α</mi><mo stretchy=\"false\">)</mo><mi>sin</mi><mo>⁡</mo><mo stretchy=\"false\">(</mo><mi>β</mi><mo stretchy=\"false\">)</mo></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column cosine left parenthesis alpha plus or minus beta right parenthesis 2nd Column equals cosine left parenthesis alpha right parenthesis cosine left parenthesis beta right parenthesis minus or plus sine left parenthesis alpha right parenthesis sine left parenthesis beta right parenthesis EndLayout","sreBraille":"⠉⠕⠎⠀⠷⠨⠁⠬⠤⠨⠃⠾⠀⠨⠅⠀⠉⠕⠎⠀⠷⠨⠁⠾⠉⠕⠎⠀⠷⠨⠃⠾⠤⠬⠎⠊⠝⠀⠷⠨⠁⠾⠎⠊⠝⠀⠷⠨⠃⠾","clearSpeak":"1 line, line 1; cosine of, open paren, alpha plus or minus beta, close paren, equals; cosine of alpha, cosine of beta; minus or plus, sine of alpha, sine of beta;","mathCatBraille":"⠉⠕⠎⠀⠷⠨⠁⠬⠤⠨⠃⠾⠀⠨⠅⠀⠉⠕⠎⠀⠷⠨⠁⠾⠉⠕⠎⠀⠷⠨⠃⠾⠤⠬⠎⠊⠝⠀⠷⠨⠁⠾⠎⠊⠝⠀⠷⠨⠃⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mfrac><mn>2</mn><mn>15</mn></mfrac></mtd><mtd><mi></mi><mo>=</mo><mn>0.133333333</mn><mo>…</mo></mtd></mtr><mtr><mtd><mfrac><mn>5</mn><mn>17</mn></mfrac></mtd><mtd><mi></mi><mo>=</mo><mn>0.</mn><munder><mn>2941176470588235</mn><mo>_</mo></munder><mn>2941176470588235</mn><munder><mn>2941176470588235</mn><mo>_</mo></munder><mn>294117647058823</mn><mo>…</mo></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column two fifteenths 2nd Column equals 0.133333333 ellipsis 2nd Row 1st Column five seventeenths 2nd Column equals 0 period ModifyingBelow 2941176470588235 With bar 2941176470588235 ModifyingBelow 2941176470588235 With bar 294117647058823 ellipsis EndLayout","sreBraille":"⠹⠆⠌⠂⠢⠼⠀⠨⠅⠀⠼⠴⠨⠂⠒⠒⠒⠒⠒⠒⠒⠒⠀⠄⠄⠄⠀⠹⠢⠌⠂⠶⠼⠀⠨⠅⠀⠼⠴⠨⠐⠆⠔⠲⠂⠂⠶⠖⠲⠶⠴⠢⠦⠦⠆⠒⠢⠩⠱⠻⠆⠔⠲⠂⠂⠶⠖⠲⠶⠴⠢⠦⠦⠆⠒⠢⠐⠆⠔⠲⠂⠂⠶⠖⠲⠶⠴⠢⠦⠦⠆⠒⠢⠩⠱⠻⠆⠔⠲⠂⠂⠶⠖⠲⠶⠴⠢⠦⠦⠆⠒⠀⠄⠄⠄","clearSpeak":"2 lines, line 1; 2 over 15, equals 0.133333333 dot dot dot; line 2; 5 over 17, equals; 0. 2941176470588235 with ¯ below, 2941176470588235 2941176470588235 with ¯ below, 294117647058823 dot dot dot;","mathCatBraille":"⠹⠆⠌⠂⠢⠼⠀⠨⠅⠀⠼⠴⠨⠂⠒⠒⠒⠒⠒⠒⠒⠒⠀⠄⠄⠄⠀⠹⠢⠌⠂⠶⠼⠀⠨⠅⠀⠼⠴⠨⠐⠐⠆⠔⠲⠂⠂⠶⠖⠲⠶⠴⠢⠦⠦⠆⠒⠢⠩⠱⠻⠆⠔⠲⠂⠂⠶⠖⠲⠶⠴⠢⠦⠦⠆⠒⠢⠐⠆⠔⠲⠂⠂⠶⠖⠲⠶⠴⠢⠦⠦⠆⠒⠢⠩⠱⠻⠆⠔⠲⠂⠂⠶⠖⠲⠶⠴⠢⠦⠦⠆⠒⠀⠄⠄⠄"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mfrac><mn>3</mn><mn>25</mn></mfrac></mtd><mtd><mi></mi><mo>=</mo><mn>0.1200000</mn><mo>⋯</mo><mo>=</mo><mn>0.119999999</mn><mo>…</mo></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mn>7</mn><mn>32</mn></mfrac></mtd><mtd><mi></mi><mo>=</mo><mo>−</mo><mn>0.2187500000</mn><mo>⋯</mo><mo>=</mo><mo>−</mo><mn>0.2187499999</mn><mo>…</mo></mtd></mtr><mtr><mtd><mfrac><mn>9</mn><mn>20</mn></mfrac></mtd><mtd><mi></mi><mo>=</mo><mn>0.45000000</mn><mo>⋯</mo><mo>=</mo><mn>0.4499999</mn><mo>…</mo></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column three twenty fifths 2nd Column equals 0.1200000 midline horizontal ellipsis equals 0.119999999 ellipsis 2nd Row 1st Column negative seven thirty seconds 2nd Column equals negative 0.2187500000 midline horizontal ellipsis equals negative 0.2187499999 ellipsis 3rd Row 1st Column nine twentieths 2nd Column equals 0.45000000 midline horizontal ellipsis equals 0.4499999 ellipsis EndLayout","sreBraille":"⠹⠒⠌⠆⠢⠼⠀⠨⠅⠀⠼⠴⠨⠂⠆⠴⠴⠴⠴⠴⠀⠄⠄⠄⠀⠀⠨⠅⠀⠼⠴⠨⠂⠂⠔⠔⠔⠔⠔⠔⠔⠀⠄⠄⠄⠀⠤⠹⠶⠌⠒⠆⠼⠀⠨⠅⠀⠤⠼⠴⠨⠆⠂⠦⠶⠢⠴⠴⠴⠴⠴⠀⠄⠄⠄⠀⠀⠨⠅⠀⠤⠼⠴⠨⠆⠂⠦⠶⠲⠔⠔⠔⠔⠔⠀⠄⠄⠄⠀⠹⠔⠌⠆⠴⠼⠀⠨⠅⠀⠼⠴⠨⠲⠢⠴⠴⠴⠴⠴⠴⠀⠄⠄⠄⠀⠀⠨⠅⠀⠼⠴⠨⠲⠲⠔⠔⠔⠔⠔⠀⠄⠄⠄","clearSpeak":"3 lines, line 1; 3 over 25, equals 0.1200000 dot dot dot, equals 0.119999999 dot dot dot; line 2; negative 7 over 32, equals, negative 0.2187500000, dot dot dot, equals, negative 0.2187499999, dot dot dot; line 3; 9 over 20, equals 0.45000000 dot dot dot, equals 0.4499999 dot dot dot;","mathCatBraille":"⠹⠒⠌⠆⠢⠼⠀⠨⠅⠀⠼⠴⠨⠂⠆⠴⠴⠴⠴⠴⠄⠄⠄⠀⠨⠅⠀⠼⠴⠨⠂⠂⠔⠔⠔⠔⠔⠔⠔⠀⠄⠄⠄⠀⠤⠹⠶⠌⠒⠆⠼⠀⠨⠅⠀⠤⠼⠴⠨⠆⠂⠦⠶⠢⠴⠴⠴⠴⠴⠄⠄⠄⠀⠨⠅⠀⠤⠼⠴⠨⠆⠂⠦⠶⠲⠔⠔⠔⠔⠔⠀⠄⠄⠄⠀⠹⠔⠌⠆⠴⠼⠀⠨⠅⠀⠼⠴⠨⠲⠢⠴⠴⠴⠴⠴⠴⠄⠄⠄⠀⠨⠅⠀⠼⠴⠨⠲⠲⠔⠔⠔⠔⠔⠀⠄⠄⠄"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi>S</mi></mtd><mtd><mi></mi><mo>=</mo><mo fence=\"false\" stretchy=\"false\">{</mo><mi>x</mi><mrow class=\"MJX-TeXAtom-ORD\"><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mtext> s.t. </mtext></mstyle></mrow><mi>x</mi><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mtext> where </mtext></mstyle><mi>k</mi><mo>∈</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"double-struck\">Z</mi></mrow><mo fence=\"false\" stretchy=\"false\">}</mo></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column upper S 2nd Column equals left brace x s period t period x equals k squared where k element of double struck upper Z right brace EndLayout","sreBraille":"⠠⠎⠀⠨⠅⠀⠨⠷⠭⠀⠎⠨⠞⠨⠀⠭⠀⠨⠅⠀⠅⠘⠆⠐⠀⠺⠓⠑⠗⠑⠀⠅⠀⠈⠑⠀⠈⠰⠠⠵⠨⠾","clearSpeak":"1 line, line 1; cap s, equals; the set x s.t. x, equals k squared where k in the integers;","mathCatBraille":"⠠⠎⠀⠨⠅⠀⠨⠷⠭⠀⠎⠸⠲⠞⠸⠲⠀⠭⠀⠨⠅⠀⠅⠘⠆⠀⠺⠓⠑⠗⠑⠀⠅⠀⠈⠑⠀⠈⠰⠠⠵⠨⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy=\"false\">]</mo><mo>=</mo><mo fence=\"false\" stretchy=\"false\">{</mo><mi>x</mi><mo>∈</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"double-struck\">R</mi></mrow><mo>:</mo><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi><mo fence=\"false\" stretchy=\"false\">}</mo></math>","mathSpeak":"left bracket a comma b right bracket equals StartSet x element of double struck upper R colon a less than or equals x less than or equals b EndSet","sreBraille":"⠈⠷⠁⠠⠀⠃⠈⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠁⠀⠐⠅⠱⠀⠭⠀⠐⠅⠱⠀⠃⠨⠾","clearSpeak":"the interval from eigh to b, including eigh and b; equals; the set of all x in the real numbers such that eigh is less than or equal to x is less than or equal to b","mathCatBraille":"⠈⠷⠁⠠⠀⠃⠈⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠀⠁⠀⠐⠅⠱⠀⠭⠀⠐⠅⠱⠀⠃⠨⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo fence=\"false\" stretchy=\"false\">{</mo><mi>x</mi><mo>∈</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"double-struck\">R</mi></mrow><mo>:</mo><mi>a</mi><mo><</mo><mi>x</mi><mo><</mo><mi>b</mi><mo fence=\"false\" stretchy=\"false\">}</mo></math>","mathSpeak":"left parenthesis a comma b right parenthesis equals StartSet x element of double struck upper R colon a less than x less than b EndSet","sreBraille":"⠷⠁⠠⠀⠃⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠁⠀⠐⠅⠀⠭⠀⠐⠅⠀⠃⠨⠾","clearSpeak":"the interval from eigh to b, not including eigh or b; equals; the set of all x in the real numbers such that eigh is less than x is less than b","mathCatBraille":"⠷⠁⠠⠀⠃⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠀⠁⠀⠐⠅⠀⠭⠀⠐⠅⠀⠃⠨⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mo stretchy=\"false\">[</mo><mi>a</mi><mo>,</mo><mi mathvariant=\"normal\">∞</mi><mo stretchy=\"false\">)</mo></mtd><mtd><mi></mi><mo>=</mo><mo fence=\"false\" stretchy=\"false\">{</mo><mi>x</mi><mo>∈</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"double-struck\">R</mi></mrow><mo>:</mo><mi>a</mi><mo>≤</mo><mi>x</mi><mo fence=\"false\" stretchy=\"false\">}</mo></mtd><mtd><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><mi mathvariant=\"normal\">∞</mi><mo stretchy=\"false\">)</mo></mtd><mtd><mi></mi><mo>=</mo><mo fence=\"false\" stretchy=\"false\">{</mo><mi>x</mi><mo>∈</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"double-struck\">R</mi></mrow><mo>:</mo><mi>a</mi><mo><</mo><mi>x</mi><mo fence=\"false\" stretchy=\"false\">}</mo></mtd></mtr><mtr><mtd><mo stretchy=\"false\">(</mo><mo>−</mo><mi mathvariant=\"normal\">∞</mi><mo>,</mo><mi>b</mi><mo stretchy=\"false\">]</mo></mtd><mtd><mi></mi><mo>=</mo><mo fence=\"false\" stretchy=\"false\">{</mo><mi>x</mi><mo>∈</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"double-struck\">R</mi></mrow><mo>:</mo><mi>x</mi><mo>≤</mo><mi>b</mi><mo fence=\"false\" stretchy=\"false\">}</mo></mtd><mtd><mo stretchy=\"false\">(</mo><mo>−</mo><mi mathvariant=\"normal\">∞</mi><mo>,</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mtd><mtd><mi></mi><mo>=</mo><mo fence=\"false\" stretchy=\"false\">{</mo><mi>x</mi><mo>∈</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"double-struck\">R</mi></mrow><mo>:</mo><mi>x</mi><mo><</mo><mi>b</mi><mo fence=\"false\" stretchy=\"false\">}</mo></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column left bracket a comma normal infinity right parenthesis 2nd Column equals StartSet x element of double struck upper R colon a less than or equals x EndSet 3rd Column left parenthesis a comma normal infinity right parenthesis 4th Column equals StartSet x element of double struck upper R colon a less than x EndSet 2nd Row 1st Column left parenthesis negative normal infinity comma b right bracket 2nd Column equals StartSet x element of double struck upper R colon x less than or equals b EndSet 3rd Column left parenthesis negative normal infinity comma b right parenthesis 4th Column equals StartSet x element of double struck upper R colon x less than b EndSet EndLayout","sreBraille":"⠈⠷⠁⠠⠀⠠⠿⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠁⠀⠐⠅⠱⠀⠭⠨⠾⠀⠷⠁⠠⠀⠠⠿⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠁⠀⠐⠅⠀⠭⠨⠾⠀⠷⠤⠠⠿⠠⠀⠃⠈⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠭⠀⠐⠅⠱⠀⠃⠨⠾⠀⠷⠤⠠⠿⠠⠀⠃⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠭⠀⠐⠅⠀⠃⠨⠾","clearSpeak":"2 lines, line 1; open bracket, eigh comma infinity, close paren, equals; the set of all x in the real numbers such that eigh is less than or equal to x, open paren eigh comma infinity, close paren, equals, the set of all x in the real numbers such that eigh is less than x; line 2; open paren, negative infinity comma b, close bracket, equals; the set of all x in the real numbers such that x is less than or equal to b, open paren, negative infinity comma b, close paren, equals, the set of all x in the real numbers such that x is less than b;","mathCatBraille":"⠈⠷⠁⠠⠀⠠⠿⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠀⠁⠀⠐⠅⠱⠀⠭⠨⠾⠀⠷⠁⠠⠀⠠⠿⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠀⠁⠀⠐⠅⠀⠭⠨⠾⠷⠤⠠⠿⠠⠀⠃⠈⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠀⠭⠀⠐⠅⠱⠀⠃⠨⠾⠀⠷⠤⠠⠿⠠⠀⠃⠾⠀⠨⠅⠀⠨⠷⠭⠀⠈⠑⠀⠈⠰⠠⠗⠸⠒⠀⠭⠀⠐⠅⠀⠃⠨⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi>A</mi><mo>∪</mo><mi>B</mi></mtd><mtd><mi></mi><mo>=</mo><mo fence=\"false\" stretchy=\"false\">{</mo><mi>x</mi><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mi>x</mi><mo>∈</mo><mi>A</mi><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mtext> or </mtext></mstyle><mi>x</mi><mo>∈</mo><mi>B</mi><mo fence=\"false\" stretchy=\"false\">}</mo></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column upper A union upper B 2nd Column equals StartSet x vertical bar x element of upper A or x element of upper B EndSet EndLayout","sreBraille":"⠠⠁⠨⠬⠠⠃⠀⠨⠅⠀⠨⠷⠭⠳⠭⠀⠈⠑⠀⠠⠁⠀⠕⠗⠀⠭⠀⠈⠑⠀⠠⠃⠨⠾","clearSpeak":"1 line, line 1; cap eigh union cap b, equals; the set x divides x in cap eigh or x in cap b;","mathCatBraille":"⠠⠁⠨⠬⠠⠃⠀⠨⠅⠀⠨⠷⠭⠀⠳⠀⠭⠀⠈⠑⠀⠠⠁⠀⠕⠗⠀⠭⠀⠈⠑⠀⠠⠃⠨⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi>A</mi><mo>∩</mo><mi>B</mi></mtd><mtd><mi></mi><mo>=</mo><mo fence=\"false\" stretchy=\"false\">{</mo><mi>x</mi><mspace width=\"thickmathspace\" /><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mspace width=\"thickmathspace\" /><mi>x</mi><mo>∈</mo><mi>A</mi><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mtext> and </mtext></mstyle><mi>x</mi><mo>∈</mo><mi>B</mi><mo fence=\"false\" stretchy=\"false\">}</mo></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column upper A intersection upper B 2nd Column equals StartSet x vertical bar x element of upper A and x element of upper B EndSet EndLayout","sreBraille":"⠠⠁⠨⠩⠠⠃⠀⠨⠅⠀⠨⠷⠭⠳⠭⠀⠈⠑⠀⠠⠁⠀⠁⠝⠙⠀⠭⠀⠈⠑⠀⠠⠃⠨⠾","clearSpeak":"1 line, line 1; cap eigh intersection cap b, equals; the set x divides x in cap eigh and x in cap b;","mathCatBraille":"⠠⠁⠨⠩⠠⠃⠀⠨⠅⠀⠨⠷⠭⠀⠳⠀⠭⠀⠈⠑⠀⠠⠁⠀⠁⠝⠙⠀⠭⠀⠈⠑⠀⠠⠃⠨⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable rowspacing=\"3pt\" columnspacing=\"1em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi>f</mi><mo>:</mo><mi>A</mi><mo stretchy=\"false\">→</mo><mi>B</mi></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row f colon upper A right arrow upper B EndLayout","sreBraille":"⠋⠸⠒⠀⠠⠁⠀⠫⠕⠀⠠⠃","clearSpeak":"1 line, line 1; f colon cap eigh right arrow cap b;","mathCatBraille":"⠋⠸⠒⠀⠠⠁⠀⠫⠕⠀⠠⠃"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mtext>range of </mtext><mi>f</mi></mtd><mtd><mi></mi><mo>=</mo><mrow><mo>{</mo><mi>b</mi><mo>∈</mo><mi>B</mi><mspace width=\"thickmathspace\" /><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mspace width=\"thickmathspace\" /><mtext>there is some </mtext><mi>a</mi><mo>∈</mo><mi>A</mi><mtext> so that </mtext><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>b</mi><mo>}</mo></mrow></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mrow><mo>{</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo>∈</mo><mi>B</mi><mspace width=\"thickmathspace\" /><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mspace width=\"thickmathspace\" /><mi>a</mi><mo>∈</mo><mi>A</mi><mo>}</mo></mrow></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column range of f 2nd Column equals StartSet b element of upper B vertical bar there is some a element of upper A so that f left parenthesis a right parenthesis equals b EndSet 2nd Row 1st Column Blank 2nd Column equals StartSet f left parenthesis a right parenthesis element of upper B vertical bar a element of upper A EndSet EndLayout","sreBraille":"⠗⠁⠝⠛⠑⠀⠕⠋⠀⠋⠀⠨⠅⠀⠨⠷⠃⠀⠈⠑⠀⠠⠃⠳⠞⠓⠑⠗⠑⠀⠊⠎⠀⠎⠕⠍⠑⠀⠁⠀⠈⠑⠀⠠⠁⠀⠎⠕⠀⠞⠓⠁⠞⠀⠋⠷⠁⠾⠀⠨⠅⠀⠃⠨⠾⠀⠀⠀⠨⠅⠀⠨⠷⠋⠷⠁⠾⠀⠈⠑⠀⠠⠃⠳⠁⠀⠈⠑⠀⠠⠁⠨⠾","clearSpeak":"2 lines, line 1; range of f, equals; the set b in, B divides there is some eigh in, cap eigh so that f of eigh; equals b; line 2; equals, the set f of eigh in B divides a in cap eigh;","mathCatBraille":"⠗⠁⠝⠛⠑⠀⠕⠋⠀⠋⠀⠨⠅⠀⠨⠷⠃⠀⠈⠑⠀⠠⠃⠀⠳⠀⠞⠓⠑⠗⠑⠀⠊⠎⠀⠎⠕⠍⠑⠀⠁⠀⠈⠑⠀⠠⠁⠀⠎⠕⠀⠞⠓⠁⠞⠀⠋⠷⠁⠾⠀⠨⠅⠀⠃⠨⠾⠀⠨⠅⠀⠨⠷⠋⠷⠁⠾⠀⠈⠑⠀⠠⠃⠀⠳⠀⠁⠀⠈⠑⠀⠠⠁⠨⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable rowspacing=\"3pt\" columnspacing=\"1em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mtext>if </mtext></mstyle><msub><mi>x</mi><mn>1</mn></msub><mo>≠</mo><msub><mi>x</mi><mn>2</mn></msub><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mtext> then </mtext></mstyle><mi>f</mi><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy=\"false\">)</mo><mo>≠</mo><mi>f</mi><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy=\"false\">)</mo></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row if x 1 not equals x 2 then f left parenthesis x 1 right parenthesis not equals f left parenthesis x 2 right parenthesis EndLayout","sreBraille":"⠊⠋⠀⠭⠂⠀⠌⠨⠅⠀⠭⠆⠀⠞⠓⠑⠝⠀⠋⠷⠭⠂⠾⠀⠌⠨⠅⠀⠋⠷⠭⠆⠾","clearSpeak":"1 line, line 1; if x sub 1, is not equal to; x sub 2 then , f of open paren x sub 1 close paren; is not equal to, f of open paren x sub 2 close paren;","mathCatBraille":"⠊⠋⠀⠭⠂⠀⠌⠨⠅⠀⠭⠆⠀⠞⠓⠑⠝⠀⠋⠷⠭⠂⠾⠀⠌⠨⠅⠀⠋⠷⠭⠆⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mi>Y</mi><mo>=</mo><msup><mi>f</mi><mrow class=\"MJX-TeXAtom-ORD\"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo><mspace width=\"thickmathspace\" /><mo stretchy=\"false\">⟺</mo><mspace width=\"thickmathspace\" /><mi>f</mi><mo stretchy=\"false\">(</mo><mi>Y</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>X</mi></math>","mathSpeak":"upper Y equals f Superscript negative 1 Baseline left parenthesis upper X right parenthesis long left right double arrow f left parenthesis upper Y right parenthesis equals upper X","sreBraille":"⠠⠽⠀⠨⠅⠀⠋⠘⠤⠂⠐⠷⠠⠭⠾⠀⠀⠋⠷⠠⠽⠾⠀⠨⠅⠀⠠⠭","clearSpeak":"cap y equals; the negative 1 power of, f of cap x times; long left right double arrow, f of cap y; equals cap x","mathCatBraille":"⠠⠽⠀⠨⠅⠀⠋⠘⠤⠂⠀⠷⠠⠭⠾⠀⠫⠪⠶⠶⠶⠕⠀⠋⠷⠠⠽⠾⠀⠨⠅⠀⠠⠭"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi>v</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mtd><mtd><mi></mi><mo>=</mo><munder><mo movablelimits=\"true\" form=\"prefix\">lim</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi>h</mi><mo stretchy=\"false\">→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>s</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>+</mo><mi>h</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>s</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mi>h</mi></mfrac><mo>=</mo><mn>9.8</mn></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column v left parenthesis 1 right parenthesis 2nd Column equals limit Underscript h right arrow 0 Endscripts StartFraction s left parenthesis 1 plus h right parenthesis minus s left parenthesis 1 right parenthesis Over h EndFraction equals 9.8 EndLayout","sreBraille":"⠧⠷⠂⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠓⠀⠫⠕⠀⠼⠴⠻⠀⠹⠎⠷⠂⠬⠓⠾⠤⠎⠷⠂⠾⠌⠓⠼⠀⠨⠅⠀⠼⠔⠨⠦","clearSpeak":"1 line, line 1; v of 1, equals; the limit as h approaches 0, of; the fraction with numerator; s, open paren 1 plus h close paren; minus s of 1; and denominator h; equals 9.8;","mathCatBraille":"⠧⠷⠂⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠓⠀⠫⠕⠀⠼⠴⠻⠀⠹⠎⠷⠂⠬⠓⠾⠤⠎⠷⠂⠾⠌⠓⠼⠀⠨⠅⠀⠼⠔⠨⠦"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><munder><mo movablelimits=\"true\" form=\"prefix\">lim</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi>x</mi><mo stretchy=\"false\">→</mo><msup><mi>a</mi><mo>+</mo></msup></mrow></munder><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mtd><mtd><mi></mi><mo>=</mo><munder><mo movablelimits=\"true\" form=\"prefix\">lim</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi>x</mi><mo stretchy=\"false\">→</mo><mi>a</mi><mo>+</mo></mrow></munder><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mo movablelimits=\"true\" form=\"prefix\">lim</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi>x</mi><mo stretchy=\"false\">↓</mo><mi>a</mi></mrow></munder><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mo movablelimits=\"true\" form=\"prefix\">lim</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi>x</mi><mo stretchy=\"false\">↘</mo><mi>a</mi></mrow></munder><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>L</mi></mtd><mtd><mtext>right-hand limit</mtext></mtd></mtr><mtr><mtd><munder><mo movablelimits=\"true\" form=\"prefix\">lim</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi>x</mi><mo stretchy=\"false\">→</mo><msup><mi>a</mi><mo>−</mo></msup></mrow></munder><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mtd><mtd><mi></mi><mo>=</mo><munder><mo movablelimits=\"true\" form=\"prefix\">lim</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi>x</mi><mo stretchy=\"false\">→</mo><mi>a</mi><mo>−</mo></mrow></munder><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mo movablelimits=\"true\" form=\"prefix\">lim</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi>x</mi><mo stretchy=\"false\">↑</mo><mi>a</mi></mrow></munder><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mo movablelimits=\"true\" form=\"prefix\">lim</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi>x</mi><mo stretchy=\"false\">↗</mo><mi>a</mi></mrow></munder><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>L</mi></mtd><mtd><mtext>left-hand limit</mtext></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column limit Underscript x right arrow a Superscript plus Baseline Endscripts f left parenthesis x right parenthesis 2nd Column equals limit Underscript x right arrow a plus Endscripts f left parenthesis x right parenthesis equals limit Underscript x down arrow a Endscripts f left parenthesis x right parenthesis equals limit Underscript x down right arrow a Endscripts f left parenthesis x right parenthesis equals upper L 3rd Column right hyphen hand limit 2nd Row 1st Column limit Underscript x right arrow a Superscript minus Baseline Endscripts f left parenthesis x right parenthesis 2nd Column equals limit Underscript x right arrow a minus Endscripts f left parenthesis x right parenthesis equals limit Underscript x up arrow a Endscripts f left parenthesis x right parenthesis equals limit Underscript x up right arrow a Endscripts f left parenthesis x right parenthesis equals upper L 3rd Column left hyphen hand limit EndLayout","sreBraille":"⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠁⠘⠬⠐⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠁⠬⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠩⠒⠒⠕⠀⠁⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠰⠒⠒⠕⠀⠁⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠠⠇⠀⠗⠊⠛⠓⠞⠤⠓⠁⠝⠙⠀⠇⠊⠍⠊⠞⠀⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠁⠘⠤⠐⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠁⠤⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠣⠒⠒⠕⠀⠁⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠘⠒⠒⠕⠀⠁⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠠⠇⠀⠇⠑⠋⠞⠤⠓⠁⠝⠙⠀⠇⠊⠍⠊⠞","clearSpeak":"2 lines, line 1; the limit as x approaches, eigh raised to the plus power, of f of x, equals, the limit as x approaches eigh plus, of f of x; equals, the limit as x downwards arrow eigh, of f of x; equals, the limit as x approaches from above eigh, of f of x; equals cap l, right-hand limit; line 2; the limit as x approaches, eigh raised to the minus power, of f of x, equals, the limit as x approaches eigh minus, of f of x; equals, the limit as x upwards arrow eigh, of f of x; equals, the limit as x approaches from below eigh, of f of x; equals cap l, left-hand limit;","mathCatBraille":"⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠁⠘⠬⠐⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠁⠬⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠩⠒⠒⠕⠀⠁⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠰⠒⠒⠕⠀⠁⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠠⠇⠀⠗⠊⠛⠓⠞⠸⠤⠓⠁⠝⠙⠀⠇⠊⠍⠊⠞⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠁⠘⠤⠐⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠁⠤⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠣⠒⠒⠕⠀⠁⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠐⠇⠊⠍⠩⠭⠀⠫⠘⠒⠒⠕⠀⠁⠻⠀⠋⠷⠭⠾⠀⠨⠅⠀⠠⠇⠀⠇⠑⠋⠞⠸⠤⠓⠁⠝⠙⠀⠇⠊⠍⠊⠞"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mtd><mtd><mi></mi><mo>=</mo><mrow><mo>{</mo><mtable columnalign=\"left left\" rowspacing=\".2em\" columnspacing=\"1em\" displaystyle=\"false\"><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mtd><mtd><mtext> if </mtext><mi>x</mi><mo>></mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext> if </mtext><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn></mtd><mtd><mtext> if </mtext><mi>x</mi><mo><</mo><mn>0</mn></mtd></mtr></mtable><mo fence=\"true\" stretchy=\"true\" symmetric=\"true\"></mo></mrow></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column f left parenthesis x right parenthesis 2nd Column equals StartLayout Enlarged left brace 1st Row 1st Column x squared plus 3 2nd Column if x greater than 0 2nd Row 1st Column 0 2nd Column if x equals 0 3rd Row 1st Column x squared minus 3 2nd Column if x less than 0 EndLayout EndLayout","sreBraille":"⠋⠷⠭⠾⠀⠨⠅⠀⠨⠠⠷⠭⠘⠆⠐⠬⠒⠀⠀⠊⠋⠀⠭⠀⠨⠂⠀⠼⠴⠀⠨⠠⠷⠼⠴⠀⠀⠊⠋⠀⠭⠀⠨⠅⠀⠼⠴⠀⠨⠠⠷⠭⠘⠆⠐⠤⠒⠀⠀⠊⠋⠀⠭⠀⠐⠅⠀⠼⠴⠐","clearSpeak":"1 line, line 1; f of x, equals; 3 cases, case 1; x squared plus 3, if x is greater than 0; case 2; 0, if x equals 0; case 3; x squared minus 3, if x is less than 0;","mathCatBraille":"⠋⠷⠭⠾⠀⠨⠅⠀⠨⠷⠭⠘⠆⠐⠬⠒⠀⠊⠋⠀⠭⠀⠨⠂⠀⠼⠴⠴⠀⠊⠋⠀⠭⠀⠨⠅⠀⠼⠴⠭⠘⠆⠐⠤⠒⠀⠊⠋⠀⠭⠀⠐⠅⠀⠼⠴"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mi>g</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>−</mo><mo stretchy=\"false\">(</mo><mi>F</mi><mo>+</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow></mtd><mtd><mi></mi><mo>=</mo><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>F</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mo stretchy=\"false\">(</mo><mi>g</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mo>≤</mo><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>F</mi><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mo>+</mo><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mi>g</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>G</mi><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column StartAbsoluteValue left parenthesis f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis right parenthesis minus left parenthesis upper F plus upper G right parenthesis EndAbsoluteValue 2nd Column equals StartAbsoluteValue left parenthesis f left parenthesis x right parenthesis minus upper F right parenthesis plus left parenthesis g left parenthesis x right parenthesis minus upper G right parenthesis EndAbsoluteValue less than or equals StartAbsoluteValue f left parenthesis x right parenthesis minus upper F EndAbsoluteValue plus StartAbsoluteValue g left parenthesis x right parenthesis minus upper G EndAbsoluteValue EndLayout","sreBraille":"⠳⠷⠋⠷⠭⠾⠬⠛⠷⠭⠾⠾⠤⠷⠠⠋⠬⠠⠛⠾⠳⠀⠨⠅⠀⠳⠷⠋⠷⠭⠾⠤⠠⠋⠾⠬⠷⠛⠷⠭⠾⠤⠠⠛⠾⠳⠀⠐⠅⠱⠀⠳⠋⠷⠭⠾⠤⠠⠋⠳⠬⠳⠛⠷⠭⠾⠤⠠⠛⠳","clearSpeak":"1 line, line 1; the absolute value of open paren, f of x plus g of x, close paren; minus, open paren cap f plus cap g, close paren, equals; the absolute value of open paren, f of x minus cap f, close paren; plus, open paren, g of x minus cap g, close paren; is less than or equal to; the absolute value of f of x minus cap f; plus, the absolute value of g of x minus cap g;","mathCatBraille":"⠳⠷⠋⠷⠭⠾⠬⠛⠷⠭⠾⠾⠤⠷⠠⠋⠬⠠⠛⠾⠳⠀⠨⠅⠀⠳⠷⠋⠷⠭⠾⠤⠠⠋⠾⠬⠷⠛⠷⠭⠾⠤⠠⠛⠾⠳⠀⠐⠅⠱⠀⠳⠋⠷⠭⠾⠤⠠⠋⠳⠬⠳⠛⠷⠭⠾⠤⠠⠛⠳"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mo>≤</mo><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mi>F</mi><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|</mo></mrow><mo>+</mo><mn>1</mn></math>","mathSpeak":"StartAbsoluteValue f left parenthesis x right parenthesis EndAbsoluteValue less than or equals StartAbsoluteValue upper F EndAbsoluteValue plus 1","sreBraille":"⠳⠋⠷⠭⠾⠳⠀⠐⠅⠱⠀⠳⠠⠋⠳⠬⠂","clearSpeak":"the absolute value of f of x; is less than or equal to, the determinant of cap f, plus 1","mathCatBraille":"⠳⠋⠷⠭⠾⠳⠀⠐⠅⠱⠀⠳⠠⠋⠳⠬⠂"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable rowspacing=\"3pt\" columnspacing=\"1em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><msup><mi>f</mi><mo>′</mo></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mspace width=\"2em\" /><mfrac><mrow><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mi>f</mi></mrow><mrow><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mi>x</mi></mrow></mfrac><mspace width=\"2em\" /><mfrac><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mrow><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mi>x</mi></mrow></mfrac><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mspace width=\"2em\" /><mrow class=\"MJX-TeXAtom-ORD\"><mover><mi>f</mi><mo>˙</mo></mover></mrow><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mspace width=\"2em\" /><mi>D</mi><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mspace width=\"2em\" /><msub><mi>D</mi><mi>x</mi></msub><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>,</mo></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row f prime left parenthesis x right parenthesis StartFraction normal d f Over normal d x EndFraction StartFraction normal d Over normal d x EndFraction f left parenthesis x right parenthesis ModifyingAbove f With dot left parenthesis x right parenthesis upper D f left parenthesis x right parenthesis upper D Subscript x Baseline f left parenthesis x right parenthesis comma EndLayout","sreBraille":"⠋⠘⠄⠐⠷⠭⠾⠹⠙⠋⠌⠙⠭⠼⠹⠙⠌⠙⠭⠼⠋⠷⠭⠾⠐⠋⠣⠻⠷⠭⠾⠠⠙⠋⠷⠭⠾⠠⠙⠰⠭⠐⠋⠷⠭⠾⠠","clearSpeak":"1 line, line 1; f prime, of; x times d f over d x, d over d x; f of; x times; f dot, of; x times D, f of, x times cap d sub x f of x; comma;","mathCatBraille":"⠋⠄⠀⠷⠭⠾⠹⠙⠋⠌⠙⠭⠼⠹⠙⠌⠙⠭⠼⠋⠀⠷⠭⠾⠐⠋⠣⠸⠲⠻⠀⠷⠭⠾⠠⠙⠋⠀⠷⠭⠾⠠⠙⠰⠭⠐⠋⠷⠭⠾⠠"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mrow><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mi>x</mi></mrow></mfrac><mi>sin</mi><mo>⁡</mo><mi>x</mi><msub><mstyle scriptlevel=\"0\"><mrow class=\"MJX-TeXAtom-ORD\"><mo maxsize=\"1.2em\" minsize=\"1.2em\">|</mo></mrow></mstyle><mrow class=\"MJX-TeXAtom-ORD\"><mi>x</mi><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mtext>.</mtext></math>","mathSpeak":"StartFraction normal d Over normal d x EndFraction sine x vertical bar Subscript x equals 0 Baseline equals 1 period","sreBraille":"⠹⠙⠌⠙⠭⠼⠎⠊⠝⠀⠭⠳⠰⠭⠀⠰⠨⠅⠀⠼⠴⠐⠨⠅⠀⠼⠂⠨","clearSpeak":"d over d x sine of x; evaluated at x equals 0; equals 1","mathCatBraille":"⠹⠙⠌⠙⠭⠼⠎⠊⠝⠀⠭⠳⠰⠭⠀⠰⠨⠅⠀⠼⠴⠀⠨⠅⠀⠼⠂"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mfrac><mrow><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mi>y</mi></mrow><mrow><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mi>x</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mrow class=\"MJX-TeXAtom-ORD\"><mi>y</mi></mrow></mrow><menclose notation=\"updiagonalstrike\"><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mi>u</mi></menclose></mfrac><mo>⋅</mo><mfrac><menclose notation=\"updiagonalstrike\"><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mrow class=\"MJX-TeXAtom-ORD\"><mi>u</mi></mrow></menclose><mrow><mrow class=\"MJX-TeXAtom-ORD\"><mi mathvariant=\"normal\">d</mi></mrow><mi>x</mi></mrow></mfrac></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column StartFraction normal d y Over normal d x EndFraction 2nd Column equals StartFraction normal d y Over CrossOut normal d u EndCrossOut EndFraction dot StartFraction CrossOut normal d u EndCrossOut Over normal d x EndFraction EndLayout","sreBraille":"⠹⠙⠽⠌⠙⠭⠼⠀⠨⠅⠀⠹⠙⠽⠌⠪⠙⠥⠻⠼⠡⠹⠪⠙⠥⠻⠌⠙⠭⠼","clearSpeak":"1 line, line 1; d y over d x, equals; the fraction with numerator d y; and denominator up diagonal, cross out, enclosing d u end enclosure; times; the fraction with numerator; up diagonal, cross out, enclosing d u end enclosure; and denominator d x;","mathCatBraille":"⠹⠙⠽⠌⠙⠭⠼⠀⠨⠅⠀⠹⠙⠽⠌⠪⠙⠥⠻⠼⠡⠹⠪⠙⠥⠻⠌⠙⠭⠼"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable rowspacing=\"3pt\" columnspacing=\"1em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi>sin</mi><mo>⁡</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mrow class=\"MJX-TeXAtom-ORD\"><mo>/</mo></mrow><mn>10</mn><mo stretchy=\"false\">)</mo><mo>≈</mo><mn>1</mn><mrow class=\"MJX-TeXAtom-ORD\"><mo>/</mo></mrow><mn>10.</mn></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row sine left parenthesis 1 divided by 10 right parenthesis almost equals 1 divided by 10 period EndLayout","sreBraille":"⠎⠊⠝⠀⠷⠂⠸⠌⠂⠴⠾⠀⠈⠱⠈⠱⠀⠼⠂⠸⠌⠂⠴⠨","clearSpeak":"1 line, line 1; sine of, open paren 1 divided by 10, close paren; is almost equal to, 1 divided by 10.;","mathCatBraille":"⠎⠊⠝⠀⠷⠂⠸⠌⠂⠴⠾⠀⠈⠱⠈⠱⠀⠼⠂⠸⠌⠂⠴⠨"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi>n</mi><mo>!</mo></mtd><mtd><mi></mi><mo>=</mo><mi>n</mi><mo>×</mo><mo stretchy=\"false\">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy=\"false\">)</mo><mo>×</mo><mo>⋯</mo><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column n factorial 2nd Column equals n times left parenthesis n minus 1 right parenthesis times midline horizontal ellipsis times 3 times 2 times 1 EndLayout","sreBraille":"⠝⠯⠀⠀⠨⠅⠀⠝⠈⠡⠷⠝⠤⠂⠾⠈⠡⠄⠄⠄⠀⠈⠡⠒⠈⠡⠆⠈⠡⠂","clearSpeak":"1 line, line 1; n factorial, equals; n times, open paren n minus 1 close paren; times; dot dot dot, times 3 times 2 times 1;","mathCatBraille":"⠝⠯⠀⠨⠅⠀⠝⠈⠡⠷⠝⠤⠂⠾⠈⠡⠄⠄⠄⠈⠡⠒⠈⠡⠆⠈⠡⠂"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi mathvariant=\"normal\">Δ</mi><mi>y</mi></mtd><mtd><mi></mi><mo>=</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo>+</mo><mi mathvariant=\"normal\">Δ</mi><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column normal upper Delta y 2nd Column equals f left parenthesis x plus normal upper Delta x right parenthesis minus f left parenthesis x right parenthesis EndLayout","sreBraille":"⠨⠠⠙⠽⠀⠨⠅⠀⠋⠷⠭⠬⠨⠠⠙⠭⠾⠤⠋⠷⠭⠾","clearSpeak":"1 line, line 1; cap delta y, equals; f of, open paren, x plus cap delta x, close paren; minus f of x;","mathCatBraille":"⠨⠠⠙⠽⠀⠨⠅⠀⠋⠷⠭⠬⠨⠠⠙⠭⠾⠤⠋⠷⠭⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><munderover><mo>∑</mo><mrow class=\"MJX-TeXAtom-ORD\"><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow class=\"MJX-TeXAtom-ORD\"><mn>10</mn></mrow></munderover><mrow><mo>[</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>]</mo></mrow></mstyle></math>","mathSpeak":"sigma summation Underscript n equals 1 Overscript 10 Endscripts left bracket StartFraction 1 Over n EndFraction minus StartFraction 1 Over n plus 1 EndFraction right bracket","sreBraille":"⠐⠨⠠⠎⠩⠝⠀⠨⠅⠀⠼⠂⠣⠂⠴⠻⠈⠷⠹⠂⠌⠝⠼⠤⠹⠂⠌⠝⠬⠂⠼⠈⠾","clearSpeak":"the sum from n equals 1 to 10 of; open bracket; 1 over n minus, the fraction with numerator 1; and denominator n plus 1; close bracket","mathCatBraille":"⠐⠨⠠⠎⠩⠝⠀⠨⠅⠀⠼⠂⠣⠂⠴⠻⠈⠷⠹⠂⠌⠝⠼⠤⠹⠂⠌⠝⠬⠂⠼⠈⠾"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" minlabelspacing=\".8em\" displaystyle=\"true\"><mtr><mtd><mi>K</mi><msub><mi>ε</mi><mi>n</mi></msub></mtd><mtd><mi></mi><mo>=</mo><msub><mi>δ</mi><mi>n</mi></msub><mo>≈</mo><msubsup><mi>δ</mi><mn>0</mn><mrow class=\"MJX-TeXAtom-ORD\"><msub><mi>F</mi><mrow class=\"MJX-TeXAtom-ORD\"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msubsup><msubsup><mi>δ</mi><mn>1</mn><mrow class=\"MJX-TeXAtom-ORD\"><msub><mi>F</mi><mi>n</mi></msub></mrow></msubsup><mo>≈</mo><msubsup><mi>δ</mi><mn>0</mn><mrow class=\"MJX-TeXAtom-ORD\"><mfrac><msup><mi>φ</mi><mrow class=\"MJX-TeXAtom-ORD\"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msqrt><mn>5</mn></msqrt></mfrac></mrow></msubsup><msubsup><mi>δ</mi><mn>1</mn><mrow class=\"MJX-TeXAtom-ORD\"><mfrac><msup><mi>φ</mi><mi>n</mi></msup><msqrt><mn>5</mn></msqrt></mfrac></mrow></msubsup><mo>=</mo><msubsup><mi>δ</mi><mn>0</mn><mrow class=\"MJX-TeXAtom-ORD\"><mfrac><mn>1</mn><mrow><msqrt><mn>5</mn></msqrt><mi>φ</mi></mrow></mfrac><mo>×</mo><msup><mi>φ</mi><mi>n</mi></msup></mrow></msubsup><msubsup><mi>δ</mi><mn>1</mn><mrow class=\"MJX-TeXAtom-ORD\"><mfrac><mn>1</mn><msqrt><mn>5</mn></msqrt></mfrac><mo>×</mo><msup><mi>φ</mi><mi>n</mi></msup></mrow></msubsup></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><msup><mi>d</mi><mrow class=\"MJX-TeXAtom-ORD\"><msup><mi>φ</mi><mi>n</mi></msup></mrow></msup><mspace width=\"2em\" /><mtext>where</mtext><mspace width=\"1em\" /><mi>d</mi><mo>=</mo><msubsup><mi>δ</mi><mn>0</mn><mrow class=\"MJX-TeXAtom-ORD\"><mfrac><mn>1</mn><mrow><msqrt><mn>5</mn></msqrt><mspace width=\"thinmathspace\" /><mi>φ</mi></mrow></mfrac></mrow></msubsup><msubsup><mi>δ</mi><mn>1</mn><mrow class=\"MJX-TeXAtom-ORD\"><mfrac><mn>1</mn><msqrt><mn>5</mn></msqrt></mfrac></mrow></msubsup></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>≈</mo><msup><mi>d</mi><mrow class=\"MJX-TeXAtom-ORD\"><msup><mn>1.6</mn><mi>n</mi></msup></mrow></msup></mtd></mtr></mtable></math>","mathSpeak":"StartLayout 1st Row 1st Column upper K epsilon Subscript n 2nd Column equals delta Subscript n Baseline almost equals delta 0 Superscript upper F Super Subscript n minus 1 Baseline delta 1 Superscript upper F Super Subscript n Baseline almost equals delta 0 Superscript StartFraction phi Super Superscript n minus 1 Superscript Over StartRoot 5 EndRoot EndFraction Baseline delta 1 Superscript StartFraction phi Super Superscript n Superscript Over StartRoot 5 EndRoot EndFraction Baseline equals delta 0 Superscript StartFraction 1 Over StartRoot 5 EndRoot phi EndFraction times phi Super Superscript n Baseline delta 1 Superscript StartFraction 1 Over StartRoot 5 EndRoot EndFraction times phi Super Superscript n Baseline 2nd Row 1st Column Blank 2nd Column equals d Superscript phi Super Superscript n Superscript Baseline where d equals delta 0 Superscript StartFraction 1 Over StartRoot 5 EndRoot phi EndFraction Baseline delta 1 Superscript StartFraction 1 Over StartRoot 5 EndRoot EndFraction Baseline 3rd Row 1st Column Blank 2nd Column almost equals d Superscript 1.6 Super Superscript n EndLayout","sreBraille":"⠠⠅⠨⠑⠰⠝⠐⠀⠨⠅⠀⠨⠙⠰⠝⠀⠈⠱⠈⠱⠀⠨⠙⠴⠘⠠⠋⠘⠰⠝⠤⠂⠘⠐⠨⠙⠂⠘⠠⠋⠘⠰⠝⠀⠈⠱⠈⠱⠀⠨⠙⠴⠘⠹⠨⠈⠋⠘⠘⠝⠤⠂⠘⠌⠜⠢⠻⠼⠐⠨⠙⠂⠘⠹⠨⠈⠋⠘⠘⠝⠘⠌⠜⠢⠻⠼⠀⠨⠅⠀⠨⠙⠴⠘⠹⠂⠌⠜⠢⠻⠨⠈⠋⠼⠈⠡⠨⠈⠋⠘⠘⠝⠘⠐⠨⠙⠂⠘⠹⠂⠌⠜⠢⠻⠼⠈⠡⠨⠈⠋⠘⠘⠝⠘⠐⠀⠀⠀⠨⠅⠀⠙⠘⠨⠈⠋⠘⠘⠝⠘⠐⠺⠓⠑⠗⠑⠙⠀⠨⠅⠀⠨⠙⠴⠘⠹⠂⠌⠜⠢⠻⠨⠈⠋⠼⠐⠨⠙⠂⠘⠹⠂⠌⠜⠢⠻⠼⠐⠀⠀⠀⠈⠱⠈⠱⠀⠙⠘⠂⠨⠖⠘⠘⠝","clearSpeak":"3 lines, line 1; cap k epsilon sub n, equals; delta sub n is almost equal to; msub of, delta comma 0 raised to the cap f sub n minus 1 end sub, power; msub of, delta comma 1 raised to the cap f sub n power; is almost equal to; msub of, delta comma 0 raised to the fraction with numerator; phi raised to the n minus 1 power; and denominator the square root of 5; power; msub of, delta comma 1 raised to the fraction with numerator; phi to the n-th power; and denominator the square root of 5; power; equals; msub of, delta comma 0 raised to the exponent, the fraction with numerator 1; and denominator the square root of 5; phi; times phi to the n-th power, end exponent; msub of, delta comma 1 raised to the exponent, the fraction with numerator 1; and denominator the square root of 5; times phi to the n-th power, end exponent; line 2; equals; d raised to the exponent, phi to the n-th power, end exponent; where d; equals; msub of, delta comma 0 raised to the fraction with numerator 1; and denominator the square root of 5; φ; power; msub of, delta comma 1 raised to the fraction with numerator 1; and denominator the square root of 5; power; line 3; is almost equal to; d raised to the exponent, 1.6 to the n-th power, end exponent;","mathCatBraille":"⠠⠅⠨⠑⠰⠝⠀⠨⠅⠀⠨⠙⠰⠝⠀⠈⠱⠈⠱⠀⠨⠙⠴⠘⠠⠋⠘⠰⠝⠤⠂⠐⠨⠙⠂⠘⠠⠋⠘⠰⠝⠀⠈⠱⠈⠱⠀⠨⠙⠴⠘⠹⠨⠈⠋⠘⠘⠝⠤⠂⠘⠌⠜⠢⠻⠼⠐⠨⠙⠂⠘⠹⠨⠈⠋⠘⠘⠝⠘⠌⠜⠢⠻⠼⠀⠨⠅⠀⠨⠙⠴⠘⠹⠂⠌⠜⠢⠻⠨⠈⠋⠼⠈⠡⠨⠈⠋⠘⠘⠝⠐⠨⠙⠂⠘⠹⠂⠌⠜⠢⠻⠼⠈⠡⠨⠈⠋⠘⠘⠝⠀⠨⠅⠀⠙⠘⠨⠈⠋⠘⠘⠝⠀⠺⠓⠑⠗⠑⠀⠙⠀⠨⠅⠀⠨⠙⠴⠘⠹⠂⠌⠜⠢⠻⠨⠈⠋⠼⠐⠨⠙⠂⠘⠹⠂⠌⠜⠢⠻⠼⠀⠈⠱⠈⠱⠀⠙⠘⠂⠨⠖⠘⠘⠝"},{"mathML":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><msubsup><mi>δ</mi><mn>0</mn><mrow class=\"MJX-TeXAtom-ORD\"><msub><mi>α</mi><mrow class=\"MJX-TeXAtom-ORD\"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msubsup><msubsup><mi>δ</mi><mn>1</mn><mrow class=\"MJX-TeXAtom-ORD\"><msub><mi>β</mi><mrow class=\"MJX-TeXAtom-ORD\"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msubsup><mo>≈</mo><msubsup><mi>δ</mi><mn>0</mn><mrow class=\"MJX-TeXAtom-ORD\"><msub><mi>α</mi><mrow class=\"MJX-TeXAtom-ORD\"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msubsup><msubsup><mi>δ</mi><mn>1</mn><mrow class=\"MJX-TeXAtom-ORD\"><msub><mi>β</mi><mrow class=\"MJX-TeXAtom-ORD\"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msubsup><msubsup><mi>δ</mi><mn>0</mn><mrow class=\"MJX-TeXAtom-ORD\"><msub><mi>α</mi><mi>n</mi></msub></mrow></msubsup><msubsup><mi>δ</mi><mn>1</mn><mrow class=\"MJX-TeXAtom-ORD\"><msub><mi>β</mi><mi>n</mi></msub></mrow></msubsup></math>","mathSpeak":"delta 0 Superscript alpha Super Subscript n plus 1 Baseline delta 1 Superscript beta Super Subscript n plus 1 Baseline almost equals delta 0 Superscript alpha Super Subscript n minus 1 Baseline delta 1 Superscript beta Super Subscript n minus 1 Baseline delta 0 Superscript alpha Super Subscript n Baseline delta 1 Superscript beta Super Subscript n","sreBraille":"⠨⠙⠴⠘⠨⠁⠘⠰⠝⠬⠂⠘⠐⠨⠙⠂⠘⠨⠃⠘⠰⠝⠬⠂⠀⠈⠱⠈⠱⠀⠨⠙⠴⠘⠨⠁⠘⠰⠝⠤⠂⠘⠐⠨⠙⠂⠘⠨⠃⠘⠰⠝⠤⠂⠘⠐⠨⠙⠴⠘⠨⠁⠘⠰⠝⠘⠐⠨⠙⠂⠘⠨⠃⠘⠰⠝","clearSpeak":"msub of, delta comma 0 raised to the alpha sub n plus 1 end sub, power; msub of, delta comma 1 raised to the beta sub n plus 1 end sub, power; is almost equal to; msub of, delta comma 0 raised to the alpha sub n minus 1 end sub, power; msub of, delta comma 1 raised to the beta sub n minus 1 end sub, power; msub of, delta comma 0 raised to the alpha sub n power; msub of, delta comma 1 raised to the beta sub n power","mathCatBraille":"⠨⠙⠴⠘⠨⠁⠘⠰⠝⠬⠂⠐⠨⠙⠂⠘⠨⠃⠘⠰⠝⠬⠂⠀⠈⠱⠈⠱⠀⠨⠙⠴⠘⠨⠁⠘⠰⠝⠤⠂⠐⠨⠙⠂⠘⠨⠃⠘⠰⠝⠤⠂⠐⠨⠙⠴⠘⠨⠁⠘⠰⠝⠐⠨⠙⠂⠘⠨⠃⠘⠰⠝"}]
const createElement = (elType, text, attributeValues, config) => {
const el = document.createElement(elType)
Object.keys(attributeValues).forEach((attribute) => {
el.setAttribute(attribute, attributeValues[attribute])
})
if (text != null && text.length>0) {
if (config != null && config.html===true) {
el.innerHTML = text
} else {
el.appendChild(document.createTextNode(text))
}
}
return el
}
const installMathJax = () => {
const mjScript = document.createElement('script')
mjScript.setAttribute('src','https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js')
mjScript.setAttribute('type','text/javascript')
mjScript.setAttribute('async','')
document.head.appendChild(mjScript)
document.getElementById('loadMathJax').disabled = true
document.getElementById('loadMathJax').setAttribute('aria-disabled','true')
document.getElementById('mjLoaded').appendChild(document.createTextNode(' (MathJax requested. To remove MathJax, reload the page.)'))
}
const formatMathML = (mml) => {
const parser = new DOMParser()
const doc = parser.parseFromString(`<!DOCTYPE math PUBLIC "-//W3C//DTD MathML 2.0//EN" "http://www.w3.org/Math/DTD/mathml2/mathml2.dtd">${mml.replaceAll('&','&').replaceAll('&','&')}`,"text/xml")
let root = null
if(doc===null || doc.documentElement == null || doc.documentElement.innerHTML.includes('<parsererror')) {
let span = document.createElement('span')
span.innerHTML = mml
if(span.children.length===1 && span.firstElementChild.tagName.toLowerCase()==='math') {
root = span.firstElementChild
}
} else {
root = doc.documentElement
}
let formattedMml = '<pre><code>'
function formatTag(node) {
let tag = `<span class=\'tagChar\'><</span><span class=\'tagName\'>${node.nodeName}</span>`
let attrLength = 0
if(node!==null && node.attributes !== null && typeof node.attributes !== 'undefined' && node.attributes.length !== null && typeof node.attributes.length !== 'undefined') {
for(let i=0;i<node.attributes.length;i++) {
let attr = node.attributes[i].name
let value = node.attributes[i].value
if(attrLength>52) {
attrLength=0
tag += `<br>${' '.repeat(node.nodeName.length+1)}`
}
if(value !== null) {
tag += ` <span class=\'tagAttrName\'>${attr}</span><span class=\'tagEqSign\'>=</span><span class=\'tagValueQuote\'>"</span><span class='tagAttrValue'>${value}</span><span class=\'tagValueQuote\'>"</span>`
attrLength += attr.length + value.length
} else {
tag += ` <span class=\'tagAttrName\'>${attr}`
attrLength += attr.length
}
}
} else {
console.log('undefined?: ')
console.log(node)
console.log('-')
}
tag += `<span class='tagChar'>${node.childNodes.length<1 ? ' /':''}></span>`
return tag
}
function formatClosingTag(node) {
let tag = `<span class='tagChar'></</span><span class='tagName'>${node.nodeName}</span><span class='tagChar'>></span>`
return tag
}
function digMath(node, level) {
if(node.nodeType===8) {
formattedMml += `<span class='mathMlComment'><--${node.textContent}--></span>`
} else if(node.nodeType===3) {
formattedMml += `<br>${' '.repeat(level*2)}<span class='tagText'>${node.data}</span>`
} else if(node.childNodes.length===1 && node.firstChild.nodeType===3){
formattedMml += `<br>${' '.repeat(level*2)}${formatTag(node)}<span class='tagText'>${node.firstChild.data}</span>${formatClosingTag(node)}`
} else if(node.childNodes.length===2 && node.childNodes[0].nodeType===3 && node.childNodes[1].nodeType===8){
formattedMml += `<br>${' '.repeat(level*2)}${formatTag(node)}<span class='tagText'>${node.childNodes[0].data}</span><span class='mathMlComment'><--${node.childNodes[1].textContent}--></span>${formatClosingTag(node)}`
} else if(node.childNodes.length===3 && node.childNodes[0].nodeType===3 && node.childNodes[1].nodeType===8 && node.childNodes[2].nodeType===3){
formattedMml += `<br>${' '.repeat(level*2)}${formatTag(node)}<span class='tagText'>${node.childNodes[0].data}</span><span class='mathMlComment'><--${node.childNodes[1].textContent}--></span><span class='tagText'>${node.childNodes[2].data}</span>${formatClosingTag(node)}`
} else if(node.childNodes.length>0) {
formattedMml += `<br>${' '.repeat(level*2)}${formatTag(node)}`
node.childNodes.forEach((node2) => {
digMath(node2, level+1)
})
formattedMml += `<br>${' '.repeat(level*2)}${formatClosingTag(node)}`
} else {
formattedMml += `<br>${' '.repeat(level*2)}${formatTag(node)}`
}
}
formattedMml += formatTag(root)
root.childNodes.forEach((n) => {
digMath(n, 1)
})
formattedMml += `<br>${formatClosingTag(root)}</code></pre>`
return formattedMml
}
const copyInnerText = (id) => {
const el = document.getElementById(id)
if(el) {
navigator.clipboard.writeText(el.innerText);
}
}
window.onload = ()=>{
const report=document.getElementById('report')
math.forEach((m,i) => {
let exampleContainer = createElement('div', null, {class:'exampleContainer'})
report.appendChild(exampleContainer)
let h3 = createElement('h3',`Example ${i+1}${(m.sreBraille==m.mathCatBraille)?'': /^\s*ERROR\s*$/.test(m.mathCatBraille) ? ' (Error)' : ' (braille differs)'}`,{class:'example', id:`ex${i+1}`})
exampleContainer.appendChild(h3)
let resultContainer = createElement('div', null, {class:'resultContainer'})
exampleContainer.appendChild(resultContainer)
let outerRenderedMathContainer = createElement('div', null, {class:'outerRenderedMathContainer'})
resultContainer.appendChild(outerRenderedMathContainer)
let h4 = createElement('h4','Rendered Math',{class:'renderedMath'})
outerRenderedMathContainer.appendChild(h4)
let renderedMathContainer = createElement('div', null, {class:'renderedMathContainer'})
outerRenderedMathContainer.appendChild(renderedMathContainer)
renderedMathContainer.innerHTML = m.mathML
let speechTextAndBrailleContainer = createElement('div', null, {class: 'speechTextAndBrailleContainer'})
outerRenderedMathContainer.appendChild(speechTextAndBrailleContainer)
h4 = createElement('h4','MathSpeak',{class:'mathSpeak'})
speechTextAndBrailleContainer.appendChild(h4)
let p = createElement('p', m.mathSpeak, {class:'mathSpeak'})
speechTextAndBrailleContainer.appendChild(p)
h4 = createElement('h4','ClearSpeak',{class:'clearSpeak'})
speechTextAndBrailleContainer.appendChild(h4)
p = createElement('p', m.clearSpeak, {class:'clearSpeak'})
speechTextAndBrailleContainer.appendChild(p)
const diff = patienceDiff(m.sreBraille.split(''), m.mathCatBraille.split(''))
h4 = createElement('h4','SRE Braille',{class:'sreBraille'})
speechTextAndBrailleContainer.appendChild(h4)
p = createElement('p', null, {class:'sreBraille braille'})
speechTextAndBrailleContainer.appendChild(p)
if(!/^\s*error\s*$/i.test(m.sreBraille)) {
let jj = 0;
for(let ii=0;ii<m.sreBraille.length;ii++) {
let classNames = ''
while(jj<diff.lines.length && diff.lines[jj].aIndex<ii) {
classNames = 'insertion'
jj++
}
if(jj<diff.lines.length && diff.lines[jj].aIndex==ii && diff.lines[jj].bIndex<0) {
classNames += ' deletion'
}
const span = createElement('span', m.sreBraille[ii], {'class': classNames})
p.appendChild(span)
jj++
}
} else {
p.appendChild(document.createTextNode(m.sreBraille))
}
h4 = createElement('h4','mathCAT Braille',{class:'mathCatBraille'})
speechTextAndBrailleContainer.appendChild(h4)
p = createElement('p', m.mathCatBraille, {class:'mathCatBraille braille'})
speechTextAndBrailleContainer.appendChild(p)
let copyBut = createElement('button', `Copy Source MathML for Example ${i+1}`, {class:'copySrc'})
copyBut.addEventListener('click', e => copyInnerText(`src${i+1}`))
speechTextAndBrailleContainer.appendChild(copyBut)
let outerMathSrcContainer = createElement('div', null, {class:'outerMathSrcContainer'})
resultContainer.appendChild(outerMathSrcContainer)
h4 = createElement('h4','Math Source',{class:'mathSrc'})
outerMathSrcContainer.appendChild(h4)
let mathSrcContainer = createElement('div', null, {class: 'mathSrcContainer', id:`src${i+1}`})
outerMathSrcContainer.appendChild(mathSrcContainer)
mathSrcContainer.innerHTML = formatMathML(m.mathML)
})
document.getElementById('loadMathJax').addEventListener('click', installMathJax)
}
})()
</script>
</head>
<body>
<header>
<h1>Braille and Math Speech Tests 0002 (Mostly from Differential Calculus)</h1>
</header>
<aside aria-describedby="vc">
<fieldset>
<legend id="vc">View Controls:</legend>
<p>By default, this page includes just the MathML for the rendered versions of the math spamples. Use the Load MathJax Version 3 button below to install MathJax if desired.</p>
<button type="button" id="loadMathJax">Load MathJax Version 3</button><span role='alert' id="mjLoaded"></span>
</fieldset>
</aside>
<main>
<div id="report">
</div>
</main>
</body>
</html>