3f:null 40:["$","title",null,{"children":"calculus | Riley | Observable"}] 46:T4ea,makeTable("circular relationships", 3, { headers: [ "name", texC`\bold{f(\theta)}`, texC`\bold{\frac{\mathrm{\bold{d}} }{\mathrm{\bold{d}} \theta} ${texBracket["["]`f(\theta)`}}`, texC`\bold{\frac{\mathrm{\bold{d}} }{\mathrm{\bold{d}} \theta} ${texBracket["["]`f^{-1}(\theta)`}}` ], rows: [ [ // sine "sine", texCB`\sin(\theta)`, texCB`\cos(\theta)`, texCB`\frac{1}{\sqrt{1-\theta^2}}` ], [ // cosine "cosine", texCB`\cos(\theta)`, texCB`-\sin(\theta)`, texCB`-\frac{1}{\sqrt{1-\theta^2}}` ], [ // tangent "tangent", texCB`\tan(\theta)`, texCB`\sec(\theta)^2`, texCB`\frac{1}{1+\theta^2}` ], [ // cotangent "cotangent", texCB`\cot(\theta)`, texCB`-\csc(\theta)^2`, texCB`-\frac{1}{1+\theta^2}` ], [ // secant "secant", texCB`\sec(\theta)`, texCB`\tan(\theta)\sec(\theta)`, texCB`\frac{1}{${texAbs`\theta`}\sqrt{\theta^2-1}}` ], [ // cosecant "cosecant", texCB`\csc(\theta)`, texCB`-\cot(\theta)\csc(\theta)`, texCB`-\frac{1}{${texAbs`\theta`}\sqrt{\theta^2-1}}` ] ] })47:T719,makeTable("hyperbolic relationships", 3, { headers: [ "name", texC`\bold{f(\theta)}`, texC`\bold{\frac{\mathrm{\bold{d}} }{\mathrm{\bold{d}} \theta} ${texBracket["["]`f(\theta)`}}`, texC`\bold{\frac{\mathrm{\bold{d}} }{\mathrm{\bold{d}} \theta} ${texBracket["["]`f^{-1}(\theta)`}}` ], rows: [ [ // hyperbolic sine "hyperbolic sine", texCB`\sinh{(\theta)} = \frac{e^\theta-e^{-\theta}}{2}`, texCB`\cosh{(\theta)}`, texCB`\frac{1}{\sqrt{\theta^2+1}}` ], [ // hyperbolic cosine "hyperbolic cosine", texCB`\cosh{(\theta)} = \frac{e^\theta+e^{-\theta}}{2}`, texCB`\sinh{(\theta)}`, texCB`\frac{1}{\sqrt{\theta^2-1}}` ], [ // hyperbolic tangent "hyperbolic tangent", // texCB`\tanh{(\theta)} = \frac{\sinh{(\theta)}}{\cosh{(\theta)}} = \frac{e^\theta-e^{-\theta}}{e^\theta+e^{-\theta}}`, texCB`\tanh{(\theta)} = \frac{\sinh{(\theta)}}{\cosh{(\theta)}}`, texCB`${texFn("sech", 2)`\theta`}`, [{[makeTable.attr]: true, rowspan: 2}, texCB`\frac{1}{1-\theta^2}`] ], [ // hyperbolic cotangent "hyperbolic cotangent", texCB`\coth{(\theta)}`, texCB`-\operatorname{csch}{(\theta)}^2`, // texCB`\frac{1}{1-\theta^2}` ], [ // hyperbolic secant "hyperbolic secant", texCB`${texFn.sech`\theta`} = \frac{1}{\cosh{(\theta)}}`, texCB`-${texFn.sech`\theta`} \cdot \tanh{(\theta)}`, texCB`\frac{-1}{\theta\sqrt{1-\theta^2}}` ], [ // hyperbolic cosecant "hyperbolic cosecant", texCB`\operatorname{csch}{(\theta)}`, texCB`-\operatorname{csch}{(\theta)} \cdot \coth{(\theta)}`, texCB`\frac{-1}{${texBracket["|"]`\theta`}\sqrt{1+\theta^2}}` ] ] })48:T44c,{ const contents = wrapHeading("alternating series", 3, wrapElm("p")`${kwd.for} the alternating series ${texCIB`${texSum.n`a_n`}`}`, wrapElm("h4")`convergence`, wrapElm("dl.theorem")( wrapElm("dt")`if`, wrapElm("dd")`${texC`${texLim.n`a_n`}=0`} ${kwd.and} the terms are non-increasing in magnitude after some value of ${texC`n`}`, wrapElm("dt.with")`then`, wrapElm("dd")`the series converges` ), wrapElm("h4")`conditional convergence`, wrapElm("dl.theorem")( wrapElm("dt.with")`if`, wrapElm("dd.with")`${texCIB`${texSum.n(texAbs`a_n`)}`} converges`, wrapElm("dt.with")`then`, wrapElm("dd")`${texCIB`${texSum.n`a_n`}`} converges`, wrapElm("dt.with")`if`, wrapElm("dd.with")`${texCIB`${texSum.n(texAbs`a_n`)}`} diverges`, wrapElm("dt.with")`but`, wrapElm("dd.with")`${texCIB`${texSum.n`a_n`}`} converges`, wrapElm("dt.with")`then`, wrapElm("dd")`${texCIB`${texSum.n`a_n`}`} converges conditionally` ) ); contents.classList.add("katex-inline-block"); return texAutoColor(contents); }49:T9a2,{ const expansion = integralApproxExpansion, f = "f", fn = texFn; function* midpoint(end) { for (let i = 0; i <= end; i++) { let sub = i == end ? "n" : i; if (i > 0) { if (i == end) { yield String.raw`\cdots`; } yield "+"; } yield fn("f")`x_{${sub}}`; } } function* trapezoid(end) { for (let i = 0; i <= end; i++) { let sub = i == end ? "n" : i; if (i > 0) { if (i == end) { yield String.raw`\cdots`; } yield "+"; } let term = fn("f")`x_{${sub}}`; yield (i == 0 || i == end) ? texFrac(term)(2) : term; } } function* Kepler(end) { if (end % 2) { end--; } for (let i = 0; i <= end; i++) { let coefficient = (i == 0 || i == end) ? "" : i % 2 ? 4 : 2; let sub = i == end ? "n" : i; if (i > 0) { if (i == end) { yield String.raw`\cdots`; } yield "+"; } yield coefficient; yield fn("f")`x_{${sub}}`; } } const wrap = document.createElement("div"); wrap.append( wrapHeading("midpoint rule", 3, texCIB`${texInt.x(texFn("f")`x`)}\approx${texBracket["["](...midpoint(expansion))}\Delta x`, wrapHeading("error", 4, texCIB`\leq {\underset{a \leq x \leq b}{\max}${texBracket("(", 2)(texBracket("|", 1)(texFn("f", 1, "″")`x`))}} \times ${texFrac`${texBracket["("]`b-a`}^3``24n^2`}` ) ), wrapHeading(`${Math.random() < 0.5 ? "trapezoid" : "trapezium"} rule`, 3, texCIB`${texInt.x(texFn("f")`x`)}\approx${texBracket["["](...trapezoid(expansion))}\Delta x`, wrapHeading("error", 4, texCIB`\leq {\underset{a \leq x \leq b}{\max}${texBracket("(", 2)(texBracket("|", 1)(texFn("f", 1, "″")`x`))}} \times ${texFrac`${texBracket["("]`b-a`}^3``12n^2`}` ) ), wrapHeading(Math.random() < 0.5 ? "Kepler’s Barrel rule (Simpson’s rule)" : wrapFrag`${wrapElm("del")`Simpson’s rule`} Kepler’s Barrel rule`, 3, texCIB`${texInt.x(texFn("f")`x`)}\approx${texBracket["["](...Kepler(expansion))}\Delta x`, (() => { let p = document.createElement("p"); p.append("This only works for even ", tex`n`, " values."); return p; })(), wrapHeading("error", 4, texCIB`\leq {\underset{a \leq x \leq b}{\max}${texBracket("(", 2)(texBracket("|", 1)(texFn("f", 1, "⁗")`x`))}} \times ${texFrac`${texBracket["("]`b-a`}^5``24n^4`}` ) ) ); return texAutoColor(wrap); }4a:Tee6,{ const wrap = document.createElement("div"), t = texCIB, // integral, square brackets, parentheses, absolute value, arbitrary functions, & natural log int = texInt, bracket = texBracket["["], paren = texBracket["("], abs = texAbs, fn = texFn, ln = fn.ln, // trig & hyperbolic trig sin = fn.sin, cos = fn.cos, tan = fn.tan, cot = fn.cot, sec = fn.sec, csc = fn.csc, sinh = fn.sinh, cosh = fn.cosh, tanh = fn.tanh, coth = fn.coth, sech = fn.sech, csch = fn.csch, intU = int.u, lnU = fn.ln`u`, // integral with u & ln(u) // trig(u) & hyperbolic trig(u) sinU = fn.sin`u`, cosU = fn.cos`u`, tanU = fn.tan`u`, cotU = fn.cot`u`, secU = fn.sec`u`, cscU = fn.csc`u`, sinhU = fn.sinh`u`, coshU = fn.cosh`u`, tanhU = fn.tanh`u`, cothU = fn.coth`u`, sechU = fn.sech`u`, cschU = fn.csch`u`; { wrap.append(htl.html.fragment`

table of integrals

In these equations, ${texC`+ C`} is elided for brevity. The equations are rewritten from the OpenStax textbook Calculus Volume 2.`); } function* bundler(...items) { it: for (let i = 0; i < items.length; i++) { let curr = items[i]; if ((typeof curr == "number" || curr instanceof Number) && items[i+1] instanceof Node) { let chunk = new DocumentFragment(); let numero = document.createElement("span"); numero.classList.add("numero"); numero.append(curr); chunk.append(numero, items[++i]); yield chunk; } else { yield curr; } } } wrap.append(...Object.entries({ "basic integrals": [ 1, t`${intU`u^n`} = \frac{u^{n+1}}{n+1}, n \neq -1`, 2, t`${intU`\frac{1}{u}`} = \ln{${paren`${abs`u`}`}}`, 3, t`${intU`e^u`} = e^u`, 4, t`${intU`a^u`} = \frac{a^u}{\ln{${paren`a`}}}`, 5, t`${intU`${sinU}`}=-${cosU}`, 6, t`${intU`${cosU}`}=${sinU}`, 7, t`${intU`${secU}^2`}=${tanU}`, 8, t`${intU`${cscU}^2`}=-${cotU}`, 9, t`${intU`${secU}${tanU}`}=${secU}`, 10, t`${intU`${cscU}${cotU}`}=-${cscU}`, 11, t`${intU`${tanU}`}=${ln`${abs`${secU}`}`}`, 12, t`${intU`${cotU}`}=${ln`${abs`${sinU}`}`}`, 13, t`${intU`${secU}`}=${ln`${abs`${secU}+${tanU}`}`}`, 14, t`${intU`${cscU}`}=${ln`${abs`${cscU}-${cotU}`}`}`, 15, t`${intU`\frac{1}{\sqrt{a^2-u^2}}`}=${sin.i`\frac{u}{a}`}`, 16, t`${intU`\frac{1}{a^2+u^2}`}=\frac{${tan.i`\frac{u}{a}`}}{a}`, 17, t`${intU`\frac{1}{u\sqrt{u^2-a^2}}`}=\frac{${sec.i`\frac{u}{a}`}}{a}`, ], "trigonometric integrals": [ 18, t`${intU`${sinU}^2`}=\frac{u}{2}-\frac{${sin`2u`}}{4}`, 19, t`${intU`${cosU}^2`}=\frac{u}{2}+\frac{${sin`2u`}}{4}`, 20, t`${intU`${tanU}^2`}=${tanU}-u`, 21, t`${intU`${cotU}^2`}=-${cotU}-u`, 22, t`${intU`${sinU}^3`}=-${cosU}\cdot\frac{2+${sinU}^2}{3}`, 23, t`${intU`${cosU}^3`}=${sinU}\cdot\frac{2+${cosU}^2}{3}`, 24, t`${intU`${tanU}^3`}=\frac{${tanU}^2}{2}+${ln`${abs`${cosU}`}`}`, 25, t`${intU`${cotU}^3`}=-\frac{${cotU}^2}{2}-${ln`${abs`${sinU}`}`}`, 26, t`${intU`${secU}^3`}=\frac{${secU}\cdot${tanU}+${ln`${abs`${secU}+${tanU}`}`}}{2}`, 27, t`${intU`${cscU}^3`}=\frac{-${cscU}\cdot${cotU}+${ln`${abs`${cscU}-${cotU}`}`}}{2}`, 28, t`${intU`${sinU}^n`}=\frac{-${sinU}^{n-1}${cosU}+${paren`n-1`}${intU`${sinU}^{n-2}`}}{n}`, 29, t`${intU`${cosU}^n`}=\frac{${cosU}^{n-1}${sinU}+${paren`n-1`}${intU`${cosU}^{n-2}`}}{n}`, 30, t`${intU`${tanU}^n`}=\frac{${tanU}^{n-1}}{n-1}-${intU`${tanU}^{n-2}`}`, 31, t`${intU`${cotU}^n`}=-\frac{${cotU}^{n-1}}{n-1}-${intU`${cotU}^{n-2}`}`, ] }).map(([title, equations]) => makeSectionCluster(title, 3, ...bundler(...equations)))); return wrap; }4b:Ta80,textAutoColors = { const wrapElement = document.createElement("div"); const styleElement = document.createElement("style"); const dispElement = document.createElement("p"); const stringCompare = Intl.Collator(undefined, { sensitivity: "case", numeric: true, caseFirst: "upper" }); { let title = document.createElement("strong"); title.append("variable colour palette"); wrapElement.append(title, dispElement, styleElement); } styleElement.searchSelector = ".katex .katex-html .mord.mathdefault[data-symbol]"; dispElement.style.lineHeight = 1; dispElement.style.color = "#FFF"; dispElement.style.display = "flex"; dispElement.style.gap = "0.75ex"; let docRoot = document.body.querySelector(".observablehq-root"); let recolor = () => { let symbols = new Set([...docRoot.querySelectorAll(styleElement.searchSelector)].flatMap(({dataset: {symbol: s}}) => (s = s?.trim()) ? [s] : []).sort(stringCompare.compare)); let colors = []; for (let i = 0; i < symbols.size; i++) { colors[i] = d3.lch(50, 75, (360 * i / symbols.size)); } let newDisp = []; let newStyle = []; for (let s of symbols) { let c = ((colors.length % 2) ? colors.shift() : colors.pop()).formatHex().toUpperCase(); let dispItem = document.createElement("span"); dispItem.append(s); dispItem.style.backgroundColor = c; dispItem.style.padding = "0.5ex"; dispItem.style.minWidth = "1em"; dispItem.style.minHeight = "1em"; dispItem.style.textAlign = "center"; dispItem.style.borderRadius = "0.5ex"; newDisp.push(dispItem); newStyle.push(`.katex .katex-html .mord.mathdefault[data-symbol="${s}"] { color: ${c}; }`); } dispElement.replaceChildren(...newDisp); styleElement.innerText = newStyle.join(" "); newStyle.length = newDisp.length = 0; }; recolor(); { let watcher = new MutationObserver(records => { for (let r of records) { if (r.type == "attributes") { return recolor(); } else { for (let collection of [r.addedNodes, r.removedNodes]) { for (let n of collection) { if (!(n instanceof Element)) { continue; } if (n.matches(styleElement.searchSelector) || n.querySelector(styleElement.searchSelector)) { return recolor(); } } } } } }); watcher.observe(docRoot, { subtree: true, childList: true, // attributeFilter: [ "data-symbol" ] }); invalidation.then(() => { watcher.disconnect(); styleElement.replaceChildren(); }); } return wrapElement; }4c:T981,texStyle = html``4d:T507,texDispError = { function texDisplayError() { return String.raw`\fcolorbox{red}{null}{ ${takeTag.concat.raw(...arguments)} }`; } Object.defineProperties(texDisplayError, { apply: { value: function applyDisplayError(text, ...objects) { if (!text) { } else if (typeof text != "string" && !(text instanceof String)) { objects.unshift(text); } for (let o of objects) { Object.defineProperty(o, "toString", { value: texDisplayError.make(text || o), enumerable: true }); } return objects.length == 1 ? objects[0] : objects; } }, make: { value: function makeDisplayError() { let text; switch (arguments.length) { case 0: text = "error"; break; case 1: { text = arguments[0]; if (typeof text == "function" || text instanceof Function) { text = text.displayName || text.name || text[Symbol.toStringTag] || text; } text = text.toString(); break; } default: text = takeTag.concat.raw(...arguments); } return () => this(text); } } }) return texDisplayError; }4e:Tbdb,texFn = { const legal = new Set(), // names allowed in shorthand (like ‘\sin’) known = new Map(), // names registered in texFn bSlash = "\\", primeChars = new Map(Object.entries({ "\\prime": ["'′ʹ", '"″ʺ〞', "‴", "⁗"], "\\backprime": ["`‵", "‶〝", "‷"] }) .flatMap(([base, items]) => items.flatMap((chars = "", count) => { let currentBase = base.repeat(++count); return [...chars].map(c => [c, currentBase]); }) )), convertPrimes = s => { for (let [find, replace] of primeChars) { s = s.replaceAll(find, replace); } return s; }; invalidation.then(() => { legal.clear(); known.clear(); primeChars.clear(); }); function texFunction(name, power = 1, decorator) { if (!name) { throw new TypeError("no name"); } name = String(name); let fn; let basic = (power == 1 || power == "") && !decorator; if (decorator && decorator.toString().trimStart().match(/^[-−]/)) { decorator = decorator.toString().trimStart(); decorator = String.raw`{}^{-} ${decorator.slice(1)}`; } let allowed; if (basic && (fn = known.get(name))) { return fn; } allowed = legal.has(name); if (!allowed) { try { tex`${bSlash}${name}`; allowed = true; } catch { allowed = false; } } // potentially escape function name allowed = allowed ? String.raw`${bSlash}${name}` : name.length == 1 ? `{ ${name} }` : String.raw`\operatorname{${name}}`; if (basic) { // define basic function fn = (...contents) => String.raw`${allowed}{${texBracket["("]`${takeTag.concat.raw(...contents)}`}}`; // define inverse function fn.i = (...contents) => String.raw`${allowed}^{{}^{-}1}{${texBracket["("]`${takeTag.concat.raw(...contents)}`}}`; // cache function known.set(name, fn); } else { power = (power == 1 || power == "") ? "" : String.raw`^{ ${power} }`; if (!decorator) { decorator = ""; } else { if ([...decorator.toString()].every(c => c == " " || primeChars.has(c))) { decorator = String.raw`^{ ${convertPrimes(decorator)} }`; } else { decorator = String.raw`^{ ${decorator} }`; } } // single-use function fn = (...contents) => String.raw`${allowed}${decorator}{ ${texBracket["("]`${takeTag.concat.raw(...contents)}`} }${power}`; } return nullToString(fn); } Object.defineProperty(texFunction, "log", { value: ((...sub) => (...contents) => String.raw`\log_{${takeTag.concat.raw(...sub)}}{ \left ( ${takeTag.concat.raw(...contents)} \right )}`), enumerable: true }); for (let name of "ln sin cos tan cot sec csc sinh cosh tanh coth sech csch".trim().split(/\s+/)) { let f = texFunction(name); name in texFunction || Object.defineProperty(texFunction, name, { value: f, enumerable: true }); } return texFunction; }4f:T6da,texBracket = { const known = Object.freeze({ pairs: new Map([ ...[..."‘’“”()<>[]||".matchAll(/[\s\S]{2}/g)].flat().map(pair => [...pair]), ["\\{", "\\}"], ["\\|", "\\|"], ...["floor", "ceil", "angle"].map(name => [`\\l${name}`, `\\r${name}`]) ].sort(([a], [b]) => a < b ? -1 : +1).flatMap(pair => [pair, [...pair].reverse()])), fromUnicode: new Map(Object.entries({ "{": "\\{", "}": "\\}", "⌊": "\\lfloor", "⌋": "\\rfloor", "⌈": "\\lceil", "⌉": "\\rceil", "<": "\\langle", "〈": "\\langle", "❬": "\\langle", "⟨": "\\langle", "〈": "\\langle", ">": "\\rangle", "〉": "\\rangle", "❭": "\\rangle", "⟩": "\\rangle", "〉": "\\rangle", "||": "\\|", "‖": "\\|", "\u23F8": "\\|" })) }); const sizes = ["big", "Big", "bigg", "Bigg"].map(s => `\\${s}`); invalidation.then(() => { known.pairs.clear(); known.fromUnicode.clear(); sizes.length = 0; }); const fun = function texBracket(left = "(", size) { for (let [u, t] of known.fromUnicode) { left = left.replaceAll(u, t); } let right = known.pairs.get(left) ?? left; if (size === undefined || size === null) { left = String.raw`\left ${left}`; right = String.raw`\right ${right}`; } else if ((--size) in sizes) { left = `${sizes[size]}${left}`; right = `${sizes[size]}${right}`; } return nullToString(((...contents) => String.raw`${left} ${takeTag.concat.raw(...contents)} ${right}`)); }; for (let left of known.pairs.keys()) { Object.defineProperty(fun, left, { value: fun(left), enumerable: true }); } return fun; }50:T482,texMaybeInfinity = { const infPos = String.raw`\infty`, infNeg = String.raw`-\infty`, setPos = new Set([Infinity, "+", ...["infty", "infinity", "∞"].flatMap(s => [s, `+${s}`]) ]), setNeg = new Set([-Infinity, "-", "−", ...["infty", "infinity", "∞"].flatMap(s => [`-${s}`, `−${s}`]) ]); function texMaybeInfinity(n, bias = 0) { (typeof bias == "string" || bias instanceof String) && ((bias = bias.valueOf()) === "+" ? 1 : (bias === "-" || bias === "−") ? -1 : 0); bias = (typeof bias == "number" || bias instanceof Number) ? bias.valueOf() : 0; (n instanceof String || n instanceof Number) && (n = n.valueOf()); if (typeof n == "number" && Number.isNaN(n)) { return bias == 0 ? 0 : bias > 0 ? infNeg : infPos; } if (n === undefined || n === null) { return bias == 0 ? 0 : bias > 0 ? infPos : infNeg; } return setPos.has(n) ? infPos : setNeg.has(n) ? infNeg : n; } Object.defineProperty(texMaybeInfinity, "range", { value: function texMaybeInfiniteRange(from, to) { return [texMaybeInfinity(from, -1), texMaybeInfinity(to, +1)]; } }) return texMaybeInfinity; }51:T1624,texNumberInput = { function rand(length) { return new Array(length).fill().map(() => Math.floor(Math.random() * 10).toFixed(0)).join(""); } function innerText(node, trim = false) { if (node instanceof CharacterData) { return (node instanceof Text) ? node.textContent : ""; } return [...node.childNodes].map(c => innerText(c, false)).join(""); } function matchMord(mord, text) { if (text == null) { throw new TypeError("no match string provided"); } if (!(mord?.classList?.contains?.("mord"))) { return false; } text = text.trim(); let part = mord.innerText.trim(); if (part == text) { return mord; } if (part == text.slice(0, part.length)) { let matches = matchMord(mord.nextElementSibling, text.slice(part.length)); if (matches) { if (Array.isArray(matches)) { matches.unshift(mord); return matches; } return [mord, matches]; } } return false; } return function texNumberInput(generator = n => `n=${n}`, [min, max] = [], {value = 0, step, validate, invalidation, keyLength = 64} = {}) { const rnd = rand(keyLength); const whole = tex`${generator(rnd)}`; whole.normalize(); const semantic = [...whole.querySelector(".katex-mathml math semantics").querySelectorAll("mn")].find(n => innerText(n, true) == rnd); if (!semantic) { throw new SyntaxError("semantic tag not found"); } const setSemantic = v => semantic.replaceChildren(v); const annotation = [...whole.querySelector(".katex-mathml math semantics").querySelectorAll("annotation")].find(n => innerText(n).includes(rnd)); if (!annotation) { throw new SyntaxError("annotation tag not found"); } let setAnnotation; { let text = innerText(annotation); let pre = text.slice(0, text.indexOf(rnd)); let post = text.slice(text.indexOf(rnd) + rnd.length); setAnnotation = v => semantic.replaceChildren(`${pre}${v}${post}`); } let texInput = [...whole.querySelector(".katex-html").querySelectorAll(".mord")].find(n => matchMord(n, rnd)); if (!texInput) { throw new SyntaxError("TeX tag not found"); } texInput = matchMord(texInput, rnd); if (Array.isArray(texInput)) { for (let mord of texInput.slice(1)) { mord.remove(); } texInput = texInput.shift(); } texInput.contentEditable = true; texInput.classList.add("tex-number-input"); const testInput = document.createElement("input"); testInput.type = "number"; min == null || min === -Infinity || (testInput.min = min); max == null || max === Infinity || (testInput.max = max); step == null || (testInput.step = step); const refresh = (v = whole.value) => { setSemantic(v); setAnnotation(v); testInput.value = v; }; inputListener: { let listener = ["input", event => { testInput.value = innerText(texInput).trim(); if (!whole.checkValidity()) { event.stopImmediatePropagation(); } }, { passive: true }]; texInput.addEventListener(...listener); invalidation?.then(() => texInput.removeEventListener(...listener)); } upDownListener: if (testInput.hasAttribute("step")) { let listener = ["keydown", e => { if (e.ctrlKey || e.altKey || e.isComposing || e.metaKey) { return; } let direction; switch (e.code) { case "ArrowUp": direction = +1; break; case "ArrowDown": direction = -1; break; default: return; } let v = whole.value; if (is.number(v, {NaN: false, infinity: false})) { let step = testInput.step; step = direction * (step ? Number.parseFloat(step) : 1); e.shiftKey && (step *= 10); if (direction < 0) { // down if (testInput.min) { let min = Number.parseFloat(testInput.min); testInput.valueAsNumber = Math.max(min, v + step); } else { testInput.valueAsNumber = v + step; } } else { // up if (testInput.max) { let min = Number.parseFloat(testInput.max); testInput.valueAsNumber = Math.min(max, v + step); } else { testInput.valueAsNumber = v + step; } } if (v != testInput.valueAsNumber) { e.preventDefault(); refresh(texInput.innerText = testInput.valueAsNumber); texInput.dispatchEvent(new Event("input", {bubbles: true, cancelable: true})); } } }, { passive: false }]; texInput.addEventListener(...listener); invalidation?.then(() => texInput.removeEventListener(...listener)); } Object.defineProperties(whole, { value: { get: function value() { return Number.parseFloat(innerText(texInput).trim()); }, set: function value(v) { texInput.replaceChildren(v); return refresh(v); }, enumerable: true }, disabled: { get: function disabled() { return !texInput.contentEditable; }, set: function disabled(d) { return texInput.contentEditable = !d; }, enumerable: true }, checkValidity: { value: function checkValidity() { refresh(); return testInput.checkValidity() && validate ? validate.call(testInput, testInput.valueAsNumber ?? testInput.value) : true; }, enumerable: true } }); whole.value = value ?? min ?? max ?? 0; return whole; } }52:Tc65,makeTable = { const symAttr = Symbol("table cell attributes"); function makeCell(cell, contents) { (cell instanceof Element) ? (cell.cloneNode(Infinity)) : (cell = document.createElement(cell)); for (let c of to.multi(contents)) { if (c == null || c === "") { continue; } if (is.in(c, symAttr)) { for (let [attr, value] of Object.entries(c)) { cell.setAttribute(attr, value); } } else { cell.append(c); } } return cell; } function makeRow({row = "tr", cell = "td"} = {}, contents) { (row instanceof Element) ? (row.cloneNode(Infinity)) : (row = document.createElement(row)); for (let c of to.multi(contents)) { row.append(makeCell(cell, c)); } return row; } function makeTable(title, { multiHead = false, headers = [], rows }) { let level = 2; if (title && isHeadingNumber(arguments[1])) { level = arguments[1]; ({ headers = [], rows } = arguments[2]); } let table = document.createElement("table"); table.classList.add("katex-inline-block"); if (multiHead) { let thead = document.createElement("thead"); for (let contents of to.multi(headers)) { thead.append(makeRow({row: "tr", cell: "th"}, contents)); } table.append(thead); } else { makeHeader: { let thead = document.createElement("thead"); thead.append((Array.isArray(headers) ? headers : [...headers]) .map(header => { let elm = document.createElement("th"); header instanceof HTMLElement && texAutoColor(header); for (let h of to.multi(header)) { if (is.in(h, symAttr)) { for (let [attr, value] of Object.entries(h)) { elm.setAttribute(attr, value); } } else { elm.append(h); // h instanceof Element && texAutoColor(h); } } return elm; }).reduce((row, header) => { row.append(header); return row; }, document.createElement("tr"))); table.append(thead); } } makeBody: { table.append((Array.isArray(rows) ? rows : [...rows]) .map(row => row.map(cell => { let cellElm = document.createElement("td"); cellElm.style.verticalAlign = "middle"; for (let c of to.multi(cell)) { if (is.in(c, symAttr)) { for (let [attr, value] of Object.entries(c)) { cellElm.setAttribute(attr, value); } } else { cellElm.append(c); // c instanceof Element && texAutoColor(c); } } return cellElm; }).reduce((rowElm, cell) => { rowElm.append(cell); return rowElm; }, document.createElement("tr"))) .reduce((tbody, row) => { tbody.append(row); return tbody; }, document.createElement("tbody"))); } return title ? wrapHeading(title, level, table) : table; } Object.defineProperty(makeTable, "attr", { value: symAttr, enumerable: true }); return makeTable; }53:T1348,wrapElm = { let BS = String.fromCodePoint(0x5C); let kebab = import("//CDN.SkyPack.dev/lodash.kebabcase@4.1.1?min"); let CSSExt = "$"; function toCSSKebab(name) { let prefix = name.slice(0, Math.max(name.indexOf(CSSExt), 0)); return `${prefix ? `-${prefix}-` : ""}${kebab(name.slice(prefix.length))}`; } function setProps(element, properties) { for (let [name, value] of is.multi(properties) ? properties : Object.entries(properties)) { element[name] = value; } } function append(element, children) { for (let c of children) { (!is.a(c, Node) && is.multi(c)) ? append(element, c) : element.append(c); } } function wrapElement(...args) { let tag, // tag name (potentially with CSS-style ID & class selectors) [default: "div"] keep, // whether to make copies of the element to re-use the output function [default: false] raw, // whether to accept string contents just like String.raw [default: true] normalize, // whether to normalize the resulting element (Node.prototype.normalize) [default: true] then, // callback called for every element returned (this = element) (signature: (element) => {}) invalidation, // promise after which the saved element is forgotten attr, // attributes (as an array of pairs or an object with own enumerable properties) prop, // properties (as an array/object) propD, // property descriptors (as property descriptor object) style, // inline CSS properties (as an array/object) styleC; // inline CSS properties in camelCase (with the form ‘moz$appearance’ for prefixed or custom properties) (as an array/object) if (is.string(args[0])) { tag = args.shift(); } if (is.object(args[0]) && !is.a(args[0], Node) && !is.a(args[0], String)) { ({tag = tag, attr, prop, propD, style, styleC, keep, raw = true, normalize = true, invalidation} = args.shift()); } let e; if (tag && tag != "div") { if (tag.includes("#") || tag.includes(".")) { let tagName = tag.slice(0, Math.max(tag.indexOf("#"), tag.indexOf("."))); tag = tag.slice(tagName.length); e = document.createElement(tagName); for (let [, type, value] of tag.matchAll(/([#.])([^#.]+)/g)) { switch (type) { case "#": e.id = value; break; case ".": e.classList.add(value); } } tag = tagName; } } e ??= document.createElement(tag ||= "div"); if (attr) { for (let a of is.multi(attr) ? attr : Object.entries(attr)) { e.setAttribute(...a); }} if (style) { for (let s of is.multi(style) ? style : Object.entries(style)) { e.style.setProperty(...s); }} if (styleC) { for (let s of is.multi(styleC) ? styleC : Object.entries(styleC)) { s = [...s]; e.style.setProperty(toCSSKebab(s.shift()), ...s); s.length = 0; }} prop && (is.multi(prop) || (prop = Object.entries(prop))); if (args.length) { append(e, args); normalize && e.normalize(); } args.length = 0; if (keep) { !invalidation && (keep instanceof Promise) && (invalidation = keep); keep = e; invalidation?.then(() => { used = true; keep = true; e = undefined; }); } else if (invalidation) { invalidation.then(() => e = undefined); } const takeTagMode = raw ? takeTag.array.raw : takeTag.array; let used; return Object.defineProperty((...children) => { if (keep) { e = keep.cloneNode(Infinity); } else if (used) { throw new ReferenceError("element already used"); } else { used = true; queueMicrotask(() => e = undefined); } prop && setProps(e, prop); propD && Object.defineProperties(e, propD); if (children.length) { append(e, takeTagMode(...children)); normalize && e.normalize(); } then && then.call(e, e); return e; }, "toString", { value: () => { throw new SyntaxError(`${wrapElement.name}: broken callback chain on <${tag}${elmShorthand(e)}> element`); }, enumerable: true, configurable: true }); } Object.defineProperty(wrapElement, "CSSExt", { enumerable: true, get: (() => CSSExt), set: v => CSSExt = String(v) }); function elmShorthand(e) { const out = [e.tagName.toLowerCase()]; e.id && out.push(`#${e.id}`); if (e.classList.length) { for (let c of e.classList) { out.push(`.${c}`); }} let name; if (name = e.getAttribute("name")) { let q = name.includes('"') ? "'" : '"'; name.endsWith(BS) && (name = `${name.slice(0, -1)}${BS.repeat(2)}`); name.replaceAll(q, `${BS}${q}`); out.push(`[name=${q}${name}${q}]`); } return out.join(""); }; kebab = (await kebab).default; return wrapElement; }54:T8a4,license = { const logo = htl.svg``; logo.style.height = "1em"; logo.style.verticalAlign = "baseline"; return htl.html`

license

Use this notebook how you want. I make no guarantees of the accuracy of these formulae; if you find an error, please comment on it! Where applicable, the contents of this notebook are licensed under the WTFPL ${logo}. The static diagram of the unit circle is from Wikimedia Commons and is a colored version of Gustavb’s original work from 2006.

`; }55:[{"id":"c9dfc237e469ea75298e9efabc69ff41136533d927549826e8987df035bcf53be1d45783ed54c514cc61b733f7503f92ef1d6193bb117d170538dc6d71f7082b","href":"https://static.observableusercontent.com/files/c9dfc237e469ea75298e9efabc69ff41136533d927549826e8987df035bcf53be1d45783ed54c514cc61b733f7503f92ef1d6193bb117d170538dc6d71f7082b","name":"unit circle@1.svg","type":"image/svg+xml","lastModified":1654356215081.972,"size":172617}] 44:{"id":"af2e374f9b420294","slug":"calculus","title":"calculus","thumbnail":"b68bf840ed72b4446c42f3dc5283107cf29d117bc2baea11bfb3f931bdb8e871","access_level":"public","create_time":"$D2022-06-01T15:10:25.381Z","update_time":"$D2023-01-20T06:01:30.955Z","view_time":null,"first_public_time":"$D2022-06-10T22:32:20.272Z","has_unpublished_edits":null,"fork":false,"stars":6,"starred":false,"owner":{"type":"team","id":"d2b7afa3a61febdf","login":"rmw4269","name":"Riley","avatar_url":"https://avatars-next.observableusercontent.com/839ad38e4ca0e32eaad7be29b5bc0c3a7f4304d5f91dd0f3642acf08015b6530","tier":"starter_2024","bio":"I enjoy spreadsheets and JavaScript way too much.","following":false},"version":2795,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"817e538440b46e8a","login":"rmw4269","name":"Riley","avatar_url":"https://avatars-next.observableusercontent.com/839ad38e4ca0e32eaad7be29b5bc0c3a7f4304d5f91dd0f3642acf08015b6530","tier":null,"bio":"I enjoy spreadsheets and JavaScript way too much.\n\ntrust://Riley.science!","following":false}],"members":[{"type":"individual","id":"817e538440b46e8a","login":"rmw4269","name":"Riley","avatar_url":"https://avatars-next.observableusercontent.com/839ad38e4ca0e32eaad7be29b5bc0c3a7f4304d5f91dd0f3642acf08015b6530","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":3,"collection":null,"type":"notebook","body":{"titling":"auto","title":"calculus","cells":[{"id":0,"value":"# calculus\n\nThis is my personal study guide for the calculus Ⅱ college course that I am taking over the summer. If you want to use this code, please pin the version or copy it; I make no stability guarantees.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":796,"value":"{\n const list = ToC({ headers: [\"h2\"], filter: [\"appendix\", \"license\", \"other resources\", \"symbols\"], invalidation });\n list.style.columns = 3;\n return list;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1608,"value":"## other resources\nI have found the [Desmos graphing calculator](//Desmos.com/calculator) extremely useful. I collated various formulae from my highschool calculus class and from my college course, and I am also working on including many of the formulae in [Calculus Volume 2](//OpenStax.org/books/calculus-volume-2) by [OpenStax](//OpenStax.org).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":599,"value":"{\n let e = document.createElement(\"div\");\n {\n let header = document.createElement(\"h2\");\n header.append(\"symbols\");\n e.append(header);\n }\n e.append(...[\n [\"abcdenxyθ\", \"variables\"],\n [\"fghuv\", \"functions\"]\n ]\n .map(([symbols, type]) => {\n let paragraph = document.createElement(\"p\");\n paragraph.append(\n symbols.length == 1 ? \"The symbol \" : \"The symbols \",\n ...[...symbols].sort().flatMap((symbol, index, array) => {\n let t = tex`${symbol}`;\n switch (array.length) {\n case 1: return t;\n case 2: return (index == 0) ? [t, \" & \"] : [t];\n default: return (index == 0) ? [t] : (index == array.length - 1) ? [\", & \", t] : [\", \", t];\n }\n }),\n symbols.length == 1 ? \" represents \" : \" represent \",\n type,\n \".\"\n );\n paragraph.normalize();\n return texAutoColor(paragraph);\n }));\n return e;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"makeSectionGrid(htl.html.fragment`the big rules`,\n { title: \"power rule\", equations: [\n texCB`${texDeriv(\"\", \"x\", \"[\")`ax^b`}=abx^{b-1}`,\n texCB`${texInt.x`x^n`}=\\frac{x^{n+1}}{n+1}+C`\n ] },\n { title: \"chain rule\", equations: [\n texCB`${texDeriv(\"\", \"x\", \"[\")`f${texBracket[\"(\"](texFn(\"g\")`x`)}`}=${texFn(\"g\", \"\", \"'\")`x`}\\cdot ${texFn(\"f\", \"\", \"'\")(texFn(\"g\")`x`)})`,\n texCB`${texDeriv(\"y\", \"x\")}=${texDeriv(\"y\", \"u\")} \\cdot ${texDeriv(\"u\", \"x\")}`\n ] },\n { title: \"product rule\", equations: [\n texCB`${texDeriv(\"\", \"x\", \"[\")`uv`} = ${texFrac`u \\cdot \\mathrm{d}v``\\mathrm{d}x`} + ${texFrac`v \\cdot \\mathrm{d}u``\\mathrm{d}x`}`,\n texCB`${texDeriv(\"\", \"x\", \"[\")`uv`} = ${texDeriv(\"\", \"x\", \"[\")`u \\cdot \\mathrm{d}v + v \\cdot \\mathrm{d}u`}`\n ] },\n { title: \"quotient rule\", equations: [\n texCB`${texDeriv(\"\", \"x\")} \\left [ \\frac{f(x)}{g(x)} \\right ] = \\frac{g(x) \\cdot f'(x) - f(x) \\cdot g'(x)}{g(x)^2}`\n ] }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1829,"value":"wrapHeading(\"integration by parts\", 2,\n texAutoColor(texIBlock`${texInt.v`u`}=uv-${texInt.u`v`}`),\n wrapElm({tag: \"ol\", prop: {start: 0}, styleC: {fontVariantNumeric: \"oldstyle-nums\"}})([\n \"logarithms\",\n \"inverse trigonometry\",\n \"algebraic\",\n \"trigonometry\",\n \"exponential\"\n ].map(([first, ...rest]) => {\n let bold = document.createElement(\"b\");\n bold.style.fontVariantCaps = \"small-caps\";\n bold.append(first);\n let item = document.createElement(\"li\");\n item.append(bold, rest.join(\"\"));\n return item;\n }))\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":815,"value":"wrapHeading(\"arc length\", 2,\n wrapElm(\"p\")`The length of the continuous curve ${texC`f(x)` ?? tex`${texFn(\"f\")`x`}`} on ${texC`x=${texBracket[\"[\"]`a,b`}`} is`,\n texCIB`${texInt.def(\"x\", \"a\", \"b\")`\\sqrt{1+f'(x)^2}`}`\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":802,"value":"## trigonometry","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":338,"value":"$46","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":630,"value":"$47","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1853,"value":"### integrating powers","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1849,"value":"texAutoColor(wrapHeading(\"Pythagorean identities\", 4,\n \"For odd powers of sine or cosine and for powers of tangent or cotangent greater than two, substitute one of the powers with a Pythagoeran identity.\",\n tex.block`${texFn.sin`x`}^2+${texFn.cos`x`}^2=1`,\n tex.block`${texFn(\"sec\", 2)`x`}=${texFn(\"tan\", 2)`x`}+1`,\n tex.block`${texFn(\"csc\", 2)`x`}=${texFn(\"cot\", 2)`x`}+1`)\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1856,"value":"texAutoColor(wrapHeading(\"even powers of sine & cosine\", 4,\n \"Reduce the power with one of the half-angle formulae.\",\n tex.block`${texFn.sin`\\theta`}^2=\\frac{1-${texFn.cos`2\\theta`}}{2}`,\n tex.block`${texFn.cos`\\theta`}^2=\\frac{1+${texFn.cos`2\\theta`}}{2}`)\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1864,"value":"texAutoColor(wrapHeading(\"secant & tangent power reduction\", 4,\n tex.block`${texInt.x(texFn(\"sec\", \"n\")`x`)}=${texFrac`${texFn(\"sec\", \"n-2\")`x`}\\cdot${texFn.tan`x`}+${texBracket[\"(\"]`n-2`}\\cdot${texInt.x(texFn(\"sec\", \"n-2\")`x`)}``n-1`}`,\n tex.block`${texInt.x(texFn(\"tan\", \"n\")`x`)}=${texFrac(texFn(\"tan\", \"n-1\")`x`)`n-1`}-${texInt.x(texFn(\"tan\", \"n-2\")`x`)}`\n))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1860,"value":"texAutoColor(wrapHeading(\"different arguments\", 4,\n \"When two multiplied trigonometric functions have different arguments, substitute.\",\n tex.block`${texFn.sin`ax`}\\cdot${texFn.sin`bx`}=${texFrac`${texFn.cos`x${texBracket[\"(\"]`a-b`}`}-${texFn.cos`x${texBracket[\"(\"]`a+b`}`}``2`}`,\n tex.block`${texFn.sin`ax`}\\cdot${texFn.cos`bx`}=${texFrac`${texFn.sin`x${texBracket[\"(\"]`a-b`}`}+${texFn.sin`x${texBracket[\"(\"]`a+b`}`}``2`}`,\n tex.block`${texFn.cos`ax`}\\cdot${texFn.cos`bx`}=${texFrac`${texFn.cos`x${texBracket[\"(\"]`a-b`}`}+${texFn.cos`x${texBracket[\"(\"]`a+b`}`}``2`}`)\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":505,"value":"makeSectionCluster(\"inverse trigonometric integrals\",\n tex.block`\\int${texBracket[\"[\"]`\\frac{1}{\\sqrt{a^2-u^2}}`}\\textrm{d}u=\\sin^{-1}${texBracket[\"(\"]`\\frac{u}{a}`}+C`,\n tex.block`\\int${texBracket[\"[\"]`\\frac{1}{a^2+u^2}`}\\textrm{d}u=${texBracket[\"(\"]`\\frac{1}{a}`}\\cdot\\tan^{-1}${texBracket[\"(\"]`\\frac{u}{a}`}+C`,\n tex.block`\\int${texBracket[\"[\"]`\\frac{1}{u\\sqrt{u^2-a^2}}`}\\textrm{d}u=${texBracket[\"(\"]`\\frac{1}{a}`}\\cdot\\sec^{-1}${texBracket[\"(\"]`\\frac{${texBracket[\"|\"]`u`}}{a}`}+C`\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":457,"value":"makeSectionCluster(\"logarithms\",\n tex.block`${texDeriv(\"\", \"x\", \"[\")(texFn.ln`u`)}=\\frac{\\mathrm{d} u}{u}`,\n tex.block`${texDeriv(\"\", \"x\", \"[\")(texFn.log`b``u`)}=\\frac{\\mathrm{d}u}{u\\cdot \\ln(b)}`,\n tex.block`\\int{${texBracket[\"[\"](texFrac(1)`x`)}\\mathrm{d}x = \\ln{${texBracket[\"(\"]`|x|`}}}+C`,\n tex.block`\\int{${texBracket[\"[\"](texFn.ln`x`)}}\\mathrm{d}x = x ${texBracket[\"(\"]`\\ln{(x)}-1`}+C`,\n tex.block`\\int{${texBracket[\"[\"](texFn.log`b``x`)}}\\mathrm{d}x = \\frac{x ${texBracket[\"(\"]`\\ln{(x)}-1`}}{\\ln{(b)}}+C`\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":203,"value":"makeSectionCluster(\"miscellaneous relationships\",\n [tex.block`${texDeriv(\"\", \"x\", \"[\")`a^x`}=a^x\\ln(a)`, \", when \", tex`a>0`],\n tex.block`${texDeriv(\"\", \"x\", \"[\")`e^x`}=e^x`,\n tex.block`${texInt.x`a^x`}=\\frac{a^x}{\\ln{(a)}}+C`,\n tex.block`\\int{${texBracket(\"[\")`e^{kx}`}}\\mathrm{d}x=\\frac{e^{kx}}{k}+C`,\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1320,"value":"## infinite series","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1322,"value":"texAutoColor(wrapHeading(wrapFrag(tex`n`, \"th term test\"), 3,\n wrapElm(\"p\")`${kwd.if} ${texIBlock`${texLim.n`a_n`}\\neq 0`} ${kwd.then} ${texIBlock`${texSum.n`a_n`}`} diverges.`\n))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2532,"value":"$48","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1341,"value":"wrapHeading(wrapFrag`${texC`p`}-series test`, 3,\n wrapElm(\"dl.theorem\")(\n wrapElm(\"dt\")`for`,\n wrapElm(\"dd\")(texCIB`${texSum()`\\frac{1}{n^p}`}`),\n wrapElm(\"dt.with\")`if`,\n wrapElm(\"dd.with\")(texC`p \\leq 1`),\n wrapElm(\"dt.with\")`then`,\n wrapElm(\"dd\")`the series ${wrapElm(\"u\")`diverges`}`,\n wrapElm(\"dt.with\")`if`,\n wrapElm(\"dd.with\")(texC`p = 1`),\n wrapElm(\"dt.with\")`then`,\n wrapElm(\"dd\")`the series is the harmonic series`,\n wrapElm(\"dt.with\")`if`,\n wrapElm(\"dd.with\")(texC`1 < p`),\n wrapElm(\"dt.with\")`then`,\n wrapElm(\"dd\")`the series ${wrapElm(\"u\")`converges`}`\n )\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1346,"value":"wrapHeading(\"integral test\", 3,\n wrapElm(\"p\")`${kwd.if} ${texCIB`f${texBracket[\"(\"]`n`}=a_n`} is continuous, positive, & decreasing, ${kwd.then} ${texCIB`${texSum.n`a_n`}`} and ${texCIB`${texInt.def.n(0, Infinity)`f(n)`}`} are either both continuous or divergent.`\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1349,"value":"wrapHeading(\"ratio test\", 3,\nwrapElm(\"dl.theorem\")(\n wrapElm(\"dt\", \"let\")(),\n wrapElm(\"dd\")(texCIB`\\rho = ${texLim.n(texBracket[\"|\"]`\\frac{a_{n+1}}{a_n}`)}`),\n wrapElm(\"dt.with\")`if`,\n wrapElm(\"dd.with\")(texC`\\rho<1`),\n wrapElm(\"dt.with\")`then`,\n wrapElm(\"dd\")`the series ${wrapElm(\"u\")`converges`} absolutely`,\n wrapElm(\"dt.with\")`if`,\n wrapElm(\"dd.with\")(texC`1<\\rho`),\n wrapElm(\"dt.with\")`then`,\n wrapElm(\"dd\")`the series ${wrapElm(\"u\")`diverges`}`\n))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1361,"value":"

direct comparison test

let

${texCIB`A=${texSum.n`a_n`}`},

${texCIB`B=${texSum.n`b_n`}`}

if

${texC`0\n

then

if: ${texC`A`} diverges
then: ${texC`B`} diverges
if: ${texC`B`} converges
then: ${texC`A`} converges

if

${texC`a_n>0\n

then

${texC`A`} and ${texC`B`} are convergent

or

${texC`A`} and ${texC`B`} are divergent

geometric series

${texIBlock`${texSum.n`ar^n`}=\\frac{a}{1-r}`}, where ${texIBlock`${texBracket[\"|\"]`r`}<1`}

\n`)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2561,"value":"

remainders

${kwd.let} ${texC`R_n`} ${kwd.be} the remainder in approximating the relevant series by the sum of its first ${texC`n`} terms

alternating series

if: ${texCIB`${texSum.n`${texBracket[\"(\"]`^{-}1`}^{n+1}a_n`}`} ${kwd.is} convergent & alternating with terms non-increasing in magnitude
then: ${texC`${texAbs`R_n`}\\leq a_n + 1`}

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1957,"value":"texAutoColor(wrapHeading(\"integral approximations\", 2,\ntexIBlock`\\Delta x=${texFrac`b-a``n`}`, \" for an integral from \", tex`a`, \" to \", tex`b`, \".\"))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1962,"value":"viewof integralApproxExpansion =\ntexAutoColor(texNumberInput(n => String.raw`n=${n}`, [2, 12], {step: 1, value: 4, invalidation})) ??\nInputs.range([2, 12], {\n label: texAutoColor(texIBlock`n=`),\n step: 1,\n validate: ({value}) => value % 1 == 0 && value >= 2\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1882,"value":"$49","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":525,"value":"unitCircle = {\n const wrap = document.createElement(\"div\");\n const header = document.createElement(\"h2\");\n const image = new DOMParser().parseFromString(await FileAttachment(\"unit circle@1.svg\").text(), \"image/svg+xml\").documentElement;\n image.style.maxWidth = \"var(--cell-width)\";\n image.style.maxHeight = `${(screen.availHeight ?? (screen.height - 64)) * 0.75}px`;\n header.append(\"unit circle\");\n wrap.append(header, image, html`

This diagram is from Wikimedia Commons and is a colored version of Gustavb’s original work from 2006.

`);\n if (wrap.children.length == 1) {\n const element = wrap.firstChild;\n element.remove();\n return element;\n }\n return wrap;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":907,"value":"$4a","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":136,"value":"$4b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2685,"value":"viewof texAutoColorCaseSensitive = Inputs.toggle({\n label: \"case sensitivity for coloring Latin variable names\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":279,"value":"## appendix","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":18,"value":"$4c","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1199,"value":"texSum = {\n function texSummation(variable = \"n\", start = \"1\", end) {\n start = texMaybeInfinity(start, 0);\n end = texMaybeInfinity(end, 1);\n return nullToString((...contents) => String.raw`\\sum_{${variable} = ${start}}^{${end}}${texBracket[\"[\"](...contents)}`);\n }\n for (let v of \"iknux\") {\n Object.defineProperty(texSummation, v, {\n value: texSummation(v),\n enumerable: true\n });\n }\n return texSummation;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1173,"value":"texLim = {\n function texLimit(variable = \"n\", end) {\n end = texMaybeInfinity(end, 1);\n return nullToString((...contents) => String.raw`\\lim\\limits_{ ${variable} \\to ${end} } ${texBracket[\"[\"](...contents)}`);\n }\n for (let v of \"iknux\") {\n Object.defineProperty(texLimit, v, {\n value: texLimit(v), enumerable: true\n })\n }\n return texLimit;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":968,"value":"texParseError = {\n try {\n tex`\\csch`;\n } catch (e) {\n e = Object.getPrototypeOf(e);\n e.name ?? Object.defineProperty(e, \"name\", {\n value: \"ParseError\",\n enumerable: true,\n configurable: true\n });\n return e;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1701,"value":"$4d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":962,"value":"$4e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":49,"value":"function texDeriv(numerator = \"\", denominator = \"\", bracket, bracketSize) {\n if (bracket || (bracketSize != null)) {\n return nullToString((...contents) => `${texDeriv(numerator, denominator)} ${texBracket(bracket, bracketSize)(...contents)}`);\n }\n return String.raw`\\frac{\\mathrm{d} ${numerator}}{\\mathrm{d} ${denominator}}`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":944,"value":"texAbs = texBracket[\"|\"]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1689,"value":"texFrac = Object.defineProperties(function texFraction() {\n return nullToString((...contents) =>\n String.raw`\\frac{ ${takeTag.concat.raw(...arguments)} }{ ${takeTag.concat.raw(...contents)} }`\n );\n}, {\n \"(\": {\n value: function texFractionWithParentheses() {\n return nullToString((...contents) =>\n texBracket[\"(\"]`\\frac{ ${takeTag.concat.raw(...arguments)} }{ ${takeTag.concat.raw(...contents)} }`\n );\n },\n enumerable: true\n },\n \"[\": {\n value: function texFractionWithBrackets() {\n return nullToString((...contents) =>\n texBracket[\"[\"]`\\frac{ ${takeTag.concat.raw(...arguments)} }{ ${takeTag.concat.raw(...contents)} }`\n );\n },\n enumerable: true\n }\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":72,"value":"$4f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1445,"value":"function texDifEval(from, to, ...contents) {\n [from, to] = texMaybeInfinity.range(from, to);\n return contents.length\n ? String.raw`{ ${texBracket[\"[\"]`${contents.join(\"\")}`} } \\right |_{ ${from} }^{ ${to} }`\n : nullToString((...contents) => String.raw`\\left . { ${texBracket[\"[\"]`${takeTag.concat.raw(...contents)}`} } \\right |_{ ${from} }^{ ${to} }`);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":915,"value":"texInt = {\n function texIntegral(variable = \"x\", bracket = \"[\") {\n return (...contents) =>\n String.raw`\\int{${texBracket(bracket)(...contents)}}\\mathrm{d}${variable}`;\n }\n function texDefiniteIntegral(variable = \"x\", from = 1, to = Infinity, bracket = \"[\") {\n [from, to] = texMaybeInfinity.range(from, to);\n // return {variable, from, to, bracket};\n return (...contents) => String.raw`\\int_{${from}}^{${to}}{${texBracket(bracket)(...contents)}}\\mathrm{d}${variable}`;\n }\n Object.defineProperty(texIntegral, \"def\", {\n value: texDefiniteIntegral,\n enumerable: true\n });\n for (let v of \"nuxyvθ\") {\n Object.defineProperty(texIntegral, v, {\n value: texIntegral(v),\n enumerable: true\n });\n Object.defineProperty(texDefiniteIntegral, v, {\n value: ((...args) => texDefiniteIntegral(v, ...args)),\n enumerable: true\n });\n }\n return texIntegral;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1236,"value":"texIBlock = function texInlineBlock (...args) {\n let e = tex.block(...args);\n if (!(e instanceof Element)) {\n let wrap = document.createElement(\"span\");\n wrap.append(e);\n e = wrap;\n }\n e.classList.add(\"katex-inline-block\");\n return e;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":121,"value":"function texAutoColor(texOutput) {\n if (typeof texOutput == \"string\" || texOutput instanceof String) { return texOutput; }\n if (!(texOutput instanceof Node) && Symbol.iterator in texOutput) {\n for (let e of texOutput) {\n texAutoColor(e);\n }\n return texOutput;\n }\n for (let e of texOutput.querySelectorAll(\".katex .katex-html .mord.mathdefault\")) {\n let name = e.innerText.trim();\n if (!name) { continue; }\n if (!texAutoColorCaseSensitive && name.match(/[A-Za-z]/)) {\n name = name.toLowerCase();\n }\n e.dataset.symbol = name;\n }\n return texOutput;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2590,"value":"texCIB = function texColoredInlineBlock(...args) {\n let out = texAutoColor(tex.block(...args));\n out.classList.add(\"katex-inline-block\");\n return out;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2588,"value":"texCB = function texColoredBlock(...args) {\n return texAutoColor(tex.block(...args));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2586,"value":"texC = function texColored(...args) {\n return texAutoColor(tex(...args));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1252,"value":"$50","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1738,"value":"nullToString = {\n const descriptor = Object.freeze({\n toString: {\n value: function toString() {\n return this();\n },\n enumerable: true\n }\n });\n return function nullToString() {\n for (let o of arguments) {\n if (typeof o == \"function\" || o instanceof Function) {\n Object.defineProperties(o, descriptor);\n } else {\n throw new TypeError(\"argument not a function\");\n }\n }\n return arguments.length == 1 ? arguments[0] : arguments;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1998,"value":"$51","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":31,"value":"import {cellWidth} from \"@rmw4269/css-cell-width\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":446,"value":"cellWidth","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":638,"value":"function makeSectionCluster(title, ...items) {\n let level = 2;\n if (title && isHeadingNumber(items[0])) {\n level = items.shift();\n }\n let list = items\n .map(item => {\n let elm = document.createElement(\"li\");\n item && (\n (typeof item == \"object\" && !(item instanceof String) && Symbol.iterator in item)\n ? elm.append(...item)\n : elm.append(item));\n texAutoColor(elm);\n return elm;\n })\n .reduce((list, item) => {\n list.append(item);\n return list;\n }, document.createElement(\"ul\"));\n list.classList.add(\"sectionCluster\", \"katex-inline-block\");\n return title ? wrapHeading(title, level, list) : list;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2752,"value":"makeTable(\"nested inverses\", 3, {\n multiHead: true,\n headers: [\n [\"outer\"],\n [\"sine\", \"cosine\", \"tangent\"]\n ],\n rows: [\n [\n \"sine\",\n texCIB`${texFn.sin(texFn.sin.i`\\theta`)}`,\n texCIB`${texFn.sin(texFn.cos.i`\\theta`)}`,\n texCIB`${texFn.sin(texFn.tan.i`\\theta`)}`\n ]\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":646,"value":"$52","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":654,"value":"function makeSectionGrid(title, ...sections) {\n let level = 2;\n if (isHeadingNumber(sections[0])) {\n level = sections.shift();\n }\n let sublevel = level + (title ? 1 : 0);\n let grid = document.createElement(\"ul\");\n grid.classList.add(\"sectionGrid\");\n grid.append(...sections\n .map(({title: subtitle = \"section\", equations = []}) => {\n let listItem = document.createElement(\"li\");\n let heading = document.createElement(`h${sublevel}`);\n heading.append(subtitle);\n listItem.append(heading, ...equations.map(e => {\n if (!(e instanceof Node) && !(typeof e == \"string\" || e instanceof String) && e !== null && e instanceof Object && Symbol.iterator in e) {\n let wrap = document.createElement(\"div\");\n wrap.append(...e);\n e = wrap;\n }\n return texAutoColor(e);\n }));\n return listItem;\n }));\n return title ? wrapHeading(title, level, grid) : grid;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2473,"value":"function wrapFrag(...children) {\n children = takeTag.array.raw(...children);\n for (let i = children.length - 1; i >= 0; i--) {\n let child = children[i];\n if (!(child instanceof Node) && is.multi(child)) {\n Array.isArray(child) || (child = [...child]);\n children.splice(i, 1, ...child);\n i += child.length;\n }\n }\n const frag = document.createDocumentFragment();\n frag.append(...children);\n return frag;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":662,"value":"function wrapHeading(title, level, ...children) {\n if (!isHeadingNumber(level)) {\n children.unshift(level);\n level = 2;\n }\n if (!title && children.length <= 1) { return children.shift(); }\n let wrap = document.createElement(\"div\");\n if (title) {\n let header = document.createElement(`h${level.toFixed(0)}`);\n header.append(title);\n wrap.append(header);\n }\n wrap.append(...children);\n return wrap;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2207,"value":"$53","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2572,"value":"kwd = {\n const kwd = wrapElm(\"span.theorem-kwd\", {keep: invalidation});\n Object.defineProperties(kwd, Object.fromEntries(\"where let be for is if then else but and\".split(/\\s+/).sort().map(word => {\n let e = wrapElm(\"span.theorem-kwd\", {keep: invalidation})(word);\n invalidation.then(() => e = undefined);\n return [word, {\n get: (() => e.cloneNode(Infinity)),\n enumerable: true\n }];\n })));\n return kwd;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":701,"value":"function isHeadingNumber(n) {\n return typeof n == \"number\" || (n instanceof Number && (n = n.valueOf())) && Number.isInteger(n) && n >= 1 && n <= 6;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":792,"value":"import {ToC} from \"@rmw4269/contents\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1044,"value":"import {takeTag} from \"@rmw4269/utilities\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1425,"value":"import {debug, is, to} from \"@rmw4269/utilities\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":833,"value":"$54","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W55","stdlib":"1"}} 45:"$44:body"