2e:null 2f:["$","title",null,{"children":"Fundamental Theorem of Calculus | Bourbaki 2.0 | Observable"}] 34:T44d,*Remark.* The two Fundamental Theorems of Calculus are both saying that differentiation and integration are inverse operations, but in different orders, yet the proofs reveal a subtle point that one of the them does not rely on the completeness of the real numbers (by way of the mean value theorem); in other words, it holds for functions that are defined on the rational numbers. To illustrate it, consider ${tex.block`f(x)=\arctan(x)`} on the "rational interval" ${tex`[0,2]\cap \mathbb{Q}`}. It is meaningful to differentiable, and then to integrate, ${tex.block`\int_0^x f'(t)\, dt = \int_0^x \frac{dt}{1+t^2}`} and you'd get back ${tex`f(x)`}. However, consider the function ${tex.block`g(x)=\begin{cases} f(x) & x<\sqrt{2}\\ f(x)+1 & x>\sqrt{2} \end{cases}`} then ${tex`g'(x)=f'(x)`} for all ${tex`x\in\mathbb{Q}`}. Integrating ${tex`g'`} gives you ${tex`f`}, not ${tex`g`}. What this means is that the Newton-Leibniz formula requires the completeness of real numbers; otherwise in evaluating the integral ${tex.block`\int_0^2 \frac{dt}{1+t^2}`} you wouldn't know to use ${tex`f`} or ${tex`g`}.35:T44d,*Remark.* A useful interpretation of the Fundamental Theorem of Calculus is as the solution to an initial value problem of the simplest differential equation: ${tex.block`\begin{cases} u'(x)=f(x), & x\in I \\ u(x_0 )=u_0 \end{cases}`} The existence of solution is given by integration: ${tex.block`u(x)=u_0 + \int_{x_0}^{x} f(t)\, dt`} while the uniquess is given by the mean value theorem. As is usual for differential equations, the existence and the uniqueness are independent, and often the uniquess can be proven first, and actually help with proving the existence. A direct generalization is the simplest initial value problem of the ${tex`n`}-th order differential equation ${tex.block`\begin{cases} u^{(n)}(x)=f(x), & x\in I \\ u^{(k)}(x_0)=u_{k}, & k=0,1,\ldots,n-1\end{cases}`} for which the (unique) solution is given by ${tex.block`u(x)=\sum_{k=0}^{n-1} \frac{u_k}{k!}(x-x_0)^k + \int_{x_0}^{x}\int_{x_0}^{x_1}\cdots\int_{x_0}^{x_{n-1}} f(x_n)\, dx_{n} dx_{n-1} \cdots dx_1`} This is the crux of [Taylor's theorem](https://observablehq.com/@neobourbaki/taylors-theorem) and Taylor series.36:T548,nFTC = function({f="f", x="x", a="a", b="b"}) { f = (f != "f") ? "\\color{red}" + f + "\\color{black}" : "f"; const formula = `f(b) = f(a) + \\int_{a}^{b} f'(x)\\,dx`; let texmd = (s)=> md(s) let replace=(s) => s.replace(/\bf\b/g, f).replace(/\bx\b/g, x).replace(/\ba\b/g, a).replace(/\bb\b/g, b) const statement = `Let $f$ be a continuously differentiable function (i.e., $f'$ exists and is continuous) on an open interval of the real line $\\RR$. Then $$ f(b) = f(a) + \\int_{a}^{b} f'(x)\\,dx $$ for any $a$ and $b$ on that interval.` const proof = `Over a small interval $[x_{i}, x_{i+1}]$, the value of $f'(x)$ does not vary a lot, so that $$ \int_{x_{i}}^{x_{i+1}} f'(t)\\,dt = f'(x_{i}) (x_{i+1}-x_{i}) = f(x_{i+1})-f(x_{i}) $$ which, when sum over the inteval, gives $$ \int_{a}^{b} f'(x)\\,dx = \\sum_i f(x_{i+1}) - f(x_{i}) = f(b) - f(a) $$ More formally, let $a=x_0 ` const proof = md` *Proof:* We shall regard the two sides as functions of ${tex`${b}`}, with ${tex`${a}`} fixed. If we can show that

(1) the two functions agree at a single point;

namely, at ${tex`${b} = ${a}`}${properties_of_integral({}).name(":")} ${tex.block`${f}(\color{red}${a}\color{black}) = ${f}(${a}) + \int_{${a}}^{\color{red}${a}\color{black}} ${f}'(${x})\,d${x}`}

(2) they are differentiable and their derivatives are equal at every point of the interval,

We need to differentiate the integral with respect to the upper limit, which is also known as the Fundamental Theorem of Calculus:
${FTC1({f:f+"'", x:b, a:a, t:"x"}).statement[0]} ${FTC1({f:f+"'", x:b, a:a, t:"x"}).proof[0]}

then the ${MVT({}).name()} implies that they must be the same function.

Denoting the difference of the two functions by ${tex`h`}, then ${tex`h'(x)=0`} by (2). Then

${tex`h`} must be a constant function.

${const_fn({f:"h"}).statement} ${const_fn({f:"h"}).proof[0]}

By (1) ${tex`h(a)=0`}, so that ${tex`h(x)=0`} for all ${tex`x`}. In particular, ${tex`h(b)=0`}.

` var thm = new Proposition("Fundamental Theorem of Calculus", [statement], [proof]) thm.name_link = "[Fundamental Theorem of Calculus](https://observablehq.com/@neobourbaki/fundamental-theorem-of-calculus)" thm.formula = formula return thm }38:T666,FTC1 = function({f="f", x="x", a="a", t="t"}) { f = (f != "f") ? red(f) : "f"; var formula = tex.block`\frac{d}{d${x}}\int_{${a}}^{${x}} ${f}(${t})\,d${t} = ${f}(${x})` var statement = md` **Theorem** (Newton-Leibniz): If ${tex`${f}`} is continuous on an open interval of the real line ${tex`\mathbb R`}, then ${formula} for fixed ${tex`${a}`} on the interval. ` let h="h" var proof = md` *Proof:* By definition of the ${differentiable({}).name_link("derivative")}, we need to prove that ${tex.block`\int_{${a}}^{${x}+${h}} ${f}(${t})\,d${t} =\int_{${a}}^{${x}} ${f}(${t})\,d${t} + ${f}(${x}) ${h} + o(${h})`} as ${tex`${h}\to 0`}, which is to say ${tex.block`\lim_{${h}\to 0} \frac{1}{${h}}\int_{${x}}^{${x}+${h}} ${f}(${t})\,d${t} = ${f}(${x})`} for fixed ${tex`${x}`}. Indeed, by ${continuous({}).name_link("continuity")} of ${tex`${f}`} (at fixed ${tex`${x}`}), for any ${tex`\epsilon>0`} we may choose ${tex`${h}`} such that ${tex.block`${f}(${x})-\epsilon<${f}(${t})<${f}(${x})+\epsilon`} for ${tex`${t}\in (${x}-${h}, ${x}+${h})`}. Integrating ${tex`${f}(${t})`} from ${tex`${x}`} to ${tex`${x}+${h}`}, we have ${tex.block`(${f}(${x})-\epsilon)${h} < \int_{${x}}^{${x}+${h}}${f}(${t})\,d${t} < (${f}(${x})+\epsilon)${h}`} by ${properties_of_integral({}).name_link}, so that ${tex.block`\int_{${x}}^{${x}+${h}} ${f}(${t})\,d${t} = ${f}(${x})${h} + o(${h})`} as claimed. ` const thm = new Proposition("Fundamental Theorem of Calculus", [statement], [proof]) thm.name_link = "[Fundamental Theorem of Calculus](https://observablehq.com/@neobourbaki/fundamental-theorem-of-calculus)" thm.formula = formula return thm }39:[] 32:{"id":"623cc1319d01845d","slug":"fundamental-theorem-of-calculus","title":"Fundamental Theorem of 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2.0","avatar_url":"https://avatars-next.observableusercontent.com/3cae296e29b8beaab579b67e329aec69f0caa397dc3c7301d4cf94d343d61212","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":{"id":"18eb3fc6a282fb11","slug":"calculus","title":"calculus","owner":{"id":"6b56d6f1a056cfea","login":"neobourbaki","name":"Bourbaki 2.0"}},"type":"notebook","body":{"title":"Fundamental Theorem of Calculus","cells":[{"id":0,"value":"md`# Fundamental Theorem of Calculus\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":517,"value":"md`\n> *Though it may not be the form that is typically seen in standard textbooks, this is arguably the most natural one of all.* `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":445,"value":"vars = ({f:\"f\", x:\"x\", a:\"a\", b:\"b\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":34,"value":"FTC(vars).statement[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":23,"value":"FTC(vars).proof[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1025,"value":"vars2 = ({f:\"f\", a:\"a\", x:\"x\", t:\"t\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":503,"value":"FTC1(vars2).statement[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":699,"value":"FTC1(vars2).proof[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":663,"value":"MVT({}).statement","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1445,"value":"The usefulness of the Fundamental Theorem of Calculus is manifest (traditionally) in the form of the **Newton-Leibniz formula**: in order to integrate a function ${tex`f`}, we first find, by whatever means, a function ${tex`F`} such that ${tex`F'=f`}, and then simply evaluate ${tex`F`} at the two endpoints of integration:\n${tex.block`\\int_a^b f(x)\\, dx = F(b)-F(a)`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1352,"value":"$34","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1400,"value":"$35","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1255,"value":"md`## Generalizations\n\nClearly, for the theorem to make sense, ${tex`f'`} needs not be continuous: it can have jump discontinuities, or certain types of singularities, within the interval of integration.\n\nIn fact, the \"right\" setting for the FTC is that ${tex`f`} be **absolutely continuous**, which means that ${tex`f'`} exists almost everywhere and that ${tex`f'\\in L^1`}, where the integral must be understood as **Lebesgue integral**.\n\nIn another direction, the generaliztion into higher dimensions is Stokes' theorem.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1055,"value":"import {continuous, differentiable, const_fn, MVT} from \"@neobourbaki/real-numbers\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1077,"value":"import {properties_of_integral} from \"@neobourbaki/integrals\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1306,"value":"url = \"https://raw.githubusercontent.com/neobourbaki/Library/main/data/A4E7D3/d.json\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1289,"value":"MVT_data = (await 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