[{"data":1,"prerenderedAt":35},["ShallowReactive",2],{"post:fundamental-theorem":3},{"meta":4,"html":17,"headings":18,"hasMath":26,"prev":27,"next":31},{"slug":5,"title":6,"created":7,"url":8,"draft":9,"categories":10,"updated":15,"description":16},"fundamental-theorem","The Fundamental Theorem of Calculus","2024-03-17T00:00:00.000Z","\u002Fpost\u002Ffundamental-theorem",false,[11],{"slug":12,"title":13,"url":14},"mathematics","Mathematics","\u002Fcategories\u002Fmathematics","2024-04-02T00:00:00.000Z","Differentiation and integration are inverse operations, made precise.","\u003Cp>The two operations of calculus are inverse to one another. Making that precise is the content\nof the fundamental theorem, and the statement is worth reading carefully\u003Csup class=\"footnote-ref\">\u003Ca href=\"#fn-1\" id=\"fnref-1\" role=\"doc-noteref\" aria-label=\"Footnote 1\">1\u003C\u002Fa>\u003C\u002Fsup>.\u003C\u002Fp>\n\u003Cp>The result connects two definitions that look nothing alike: the derivative as a limit of\ndifference quotients, and the integral as a limit of Riemann sums.\u003C\u002Fp>\n\u003Ch2 id=\"the-statement\">The statement\u003Ca class=\"heading-anchor\" href=\"#the-statement\" aria-hidden=\"true\" tabindex=\"-1\">#\u003C\u002Fa>\u003C\u002Fh2>\n\u003Cdiv class=\"env env-theorem\" id=\"thm:ftc\">\n\u003Cdiv class=\"env-head\">\u003Cspan class=\"env-name\">Theorem\u003C\u002Fspan> \u003Cspan class=\"env-number\">1.1\u003C\u002Fspan> \u003Cspan class=\"env-title\">(Fundamental Theorem of Calculus)\u003C\u002Fspan>\u003C\u002Fdiv>\n\u003Cdiv class=\"env-body\">\n\u003Cp>Let \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"f\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"f\">\u003Cmjx-c class=\"mjx-c1D453\">𝑓\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan> be continuous on \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"[a,b]\" class=\"NCM-N\">\u003Cmjx-mo data-latex=\"[\">\u003Cmjx-c class=\"mjx-c5B\">[\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"a\">\u003Cmjx-c class=\"mjx-c1D44E\">𝑎\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\",\">\u003Cmjx-c class=\"mjx-c2C\">,\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi space=\"2\" data-latex=\"b\">\u003Cmjx-c class=\"mjx-c1D44F\">𝑏\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"]\">\u003Cmjx-c class=\"mjx-c5D\">]\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan> and define \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"F(x) = \\int_a^x f(t)\\,dt\" breakable=\"true\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"F\">\u003Cmjx-c class=\"mjx-c1D439\">𝐹\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-break size=\"4\"> \u003C\u002Fmjx-break>\u003Cmjx-mo data-latex=\"=\">\u003Cmjx-c class=\"mjx-c3D\">=\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-msubsup space=\"4\" data-latex=\"\\int_a^x\">\u003Cmjx-mo data-latex=\"\\int\">\u003Cmjx-c class=\"mjx-c222B\">∫\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-script style=\"vertical-align: -0.341em; margin-left: -0.12em;\">\u003Cmjx-mi size=\"s\" data-latex=\"x\" style=\"margin-left: 0.249em;\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-spacer style=\"margin-top: 0.553em;\">\u003C\u002Fmjx-spacer>\u003Cmjx-mi size=\"s\" data-latex=\"a\">\u003Cmjx-c class=\"mjx-c1D44E\">𝑎\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-script>\u003C\u002Fmjx-msubsup>\u003Cmjx-mi space=\"2\" data-latex=\"f\">\u003Cmjx-c class=\"mjx-c1D453\">𝑓\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"t\">\u003Cmjx-c class=\"mjx-c1D461\">𝑡\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-break size=\"0\"> \u003C\u002Fmjx-break>\u003Cmjx-mspace data-latex=\"\\,\" style=\"width: 0.167em;\">\u003C\u002Fmjx-mspace>\u003Cmjx-mi data-latex=\"d\">\u003Cmjx-c class=\"mjx-c1D451\">𝑑\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mi data-latex=\"t\">\u003Cmjx-c class=\"mjx-c1D461\">𝑡\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan>. Then \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"F\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"F\">\u003Cmjx-c class=\"mjx-c1D439\">𝐹\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan> is\ndifferentiable on \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"(a,b)\" class=\"NCM-N\">\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"a\">\u003Cmjx-c class=\"mjx-c1D44E\">𝑎\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\",\">\u003Cmjx-c class=\"mjx-c2C\">,\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi space=\"2\" data-latex=\"b\">\u003Cmjx-c class=\"mjx-c1D44F\">𝑏\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan> and \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"F'(x) = f(x)\" breakable=\"true\" class=\"NCM-N\">\u003Cmjx-msup>\u003Cmjx-mi data-latex=\"F\">\u003Cmjx-c class=\"mjx-c1D439\">𝐹\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-script style=\"vertical-align: 0.363em; margin-left: 0.055em;\">\u003Cmjx-mo size=\"s\" data-latex=\"'\">\u003Cmjx-c class=\"mjx-c2032 NCM-V\">′\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003C\u002Fmjx-script>\u003C\u002Fmjx-msup>\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-break size=\"4\"> \u003C\u002Fmjx-break>\u003Cmjx-mo data-latex=\"=\">\u003Cmjx-c class=\"mjx-c3D\">=\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi space=\"4\" data-latex=\"f\">\u003Cmjx-c class=\"mjx-c1D453\">𝑓\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan>.\u003C\u002Fp>\n\u003C\u002Fdiv>\n\u003C\u002Fdiv>\n\u003Cdiv class=\"env env-proof\" id=\"env-u1\">\n\u003Cdiv class=\"env-head\">\u003Cspan class=\"env-name\">Proof\u003C\u002Fspan>\u003C\u002Fdiv>\n\u003Cdiv class=\"env-body\">\n\u003Cp>Fix \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"x \\in (a,b)\" breakable=\"true\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-break size=\"4\"> \u003C\u002Fmjx-break>\u003Cmjx-mo data-latex=\"\\in\">\u003Cmjx-c class=\"mjx-c2208\">∈\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mo space=\"4\" data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"a\">\u003Cmjx-c class=\"mjx-c1D44E\">𝑎\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\",\">\u003Cmjx-c class=\"mjx-c2C\">,\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi space=\"2\" data-latex=\"b\">\u003Cmjx-c class=\"mjx-c1D44F\">𝑏\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan> and let \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"h \\neq 0\" breakable=\"true\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"h\">\u003Cmjx-c class=\"mjx-c210E\">ℎ\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-break size=\"4\"> \u003C\u002Fmjx-break>\u003Cmjx-mo data-latex=\"\\neq\">\u003Cmjx-c class=\"mjx-c2260\">≠\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mn space=\"4\" data-latex=\"0\">\u003Cmjx-c class=\"mjx-c30\">0\u003C\u002Fmjx-c>\u003C\u002Fmjx-mn>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan> be small enough that \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"x + h \\in (a,b)\" breakable=\"true\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-break size=\"3\"> \u003C\u002Fmjx-break>\u003Cmjx-mo data-latex=\"+\">\u003Cmjx-c class=\"mjx-c2B\">+\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi space=\"3\" data-latex=\"h\">\u003Cmjx-c class=\"mjx-c210E\">ℎ\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-break size=\"4\"> \u003C\u002Fmjx-break>\u003Cmjx-mo data-latex=\"\\in\">\u003Cmjx-c class=\"mjx-c2208\">∈\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mo space=\"4\" data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"a\">\u003Cmjx-c class=\"mjx-c1D44E\">𝑎\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\",\">\u003Cmjx-c class=\"mjx-c2C\">,\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi space=\"2\" data-latex=\"b\">\u003Cmjx-c class=\"mjx-c1D44F\">𝑏\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan>. Then\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%; min-width: 20.029em;\" display=\"true\" width=\"full\">\u003Cmjx-math data-latex=\"\\begin{equation}\\label{eq:diffquot}\n\\frac{F(x+h) - F(x)}{h} = \\frac{1}{h}\\int_x^{x+h} f(t)\\,dt.\n\\end{equation}\" width=\"full\" display=\"true\" class=\"NCM-N\">\u003Cmjx-mtable style=\"min-width: 20.029em;\" data-latex=\"\\begin{equation}\\label{eq:diffquot}\n\\frac{F(x+h) - F(x)}{h} = \\frac{1}{h}\\int_x^{x+h} f(t)\\,dt.\n\\end{equation}\" width=\"full\" side=\"right\">\u003Cmjx-table style=\"width: auto; min-width: 15.873em; margin: 0 2.078em;\">\u003Cmjx-itable width=\"full\">\u003Cmjx-mlabeledtr style=\"height: NaNem;\">\u003Cmjx-mtd>\u003Cmjx-mfrac data-latex=\"\\frac{F(x+h) - F(x)}{h}\">\u003Cmjx-frac type=\"d\">\u003Cmjx-num>\u003Cmjx-nstrut type=\"d\">\u003C\u002Fmjx-nstrut>\u003Cmjx-mrow data-latex=\"F(x+h) - F(x)\">\u003Cmjx-mi data-latex=\"F\">\u003Cmjx-c class=\"mjx-c1D439\">𝐹\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo space=\"3\" data-latex=\"+\">\u003Cmjx-c class=\"mjx-c2B\">+\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi space=\"3\" data-latex=\"h\">\u003Cmjx-c class=\"mjx-c210E\">ℎ\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mo space=\"3\" data-latex=\"-\">\u003Cmjx-c class=\"mjx-c2212\">−\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi space=\"3\" data-latex=\"F\">\u003Cmjx-c class=\"mjx-c1D439\">𝐹\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003C\u002Fmjx-mrow>\u003C\u002Fmjx-num>\u003Cmjx-dbox>\u003Cmjx-dtable>\u003Cmjx-line type=\"d\">\u003C\u002Fmjx-line>\u003Cmjx-row>\u003Cmjx-den>\u003Cmjx-dstrut type=\"d\">\u003C\u002Fmjx-dstrut>\u003Cmjx-mi data-latex=\"h\">\u003Cmjx-c class=\"mjx-c210E\">ℎ\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-den>\u003C\u002Fmjx-row>\u003C\u002Fmjx-dtable>\u003C\u002Fmjx-dbox>\u003C\u002Fmjx-frac>\u003C\u002Fmjx-mfrac>\u003Cmjx-mo space=\"4\" data-latex=\"=\">\u003Cmjx-c class=\"mjx-c3D\">=\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mfrac space=\"4\" data-latex=\"\\frac{1}{h}\">\u003Cmjx-frac type=\"d\">\u003Cmjx-num>\u003Cmjx-nstrut type=\"d\">\u003C\u002Fmjx-nstrut>\u003Cmjx-mn data-latex=\"1\">\u003Cmjx-c class=\"mjx-c31\">1\u003C\u002Fmjx-c>\u003C\u002Fmjx-mn>\u003C\u002Fmjx-num>\u003Cmjx-dbox>\u003Cmjx-dtable>\u003Cmjx-line type=\"d\">\u003C\u002Fmjx-line>\u003Cmjx-row>\u003Cmjx-den>\u003Cmjx-dstrut type=\"d\">\u003C\u002Fmjx-dstrut>\u003Cmjx-mi data-latex=\"h\">\u003Cmjx-c class=\"mjx-c210E\">ℎ\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-den>\u003C\u002Fmjx-row>\u003C\u002Fmjx-dtable>\u003C\u002Fmjx-dbox>\u003C\u002Fmjx-frac>\u003C\u002Fmjx-mfrac>\u003Cmjx-msubsup space=\"2\" data-latex=\"\\int_x^{x+h}\">\u003Cmjx-mo data-latex=\"\\int\">\u003Cmjx-c class=\"mjx-c222B NCM-LO\">∫\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-script style=\"vertical-align: -0.896em; margin-left: -0.33em;\">\u003Cmjx-TeXAtom size=\"s\" data-latex=\"{x+h}\" texclass=\"ORD\" style=\"margin-left: 0.561em;\">\u003Cmjx-mi data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"+\">\u003Cmjx-c class=\"mjx-c2B\">+\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"h\">\u003Cmjx-c class=\"mjx-c210E\">ℎ\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-TeXAtom>\u003Cmjx-spacer style=\"margin-top: 1.613em;\">\u003C\u002Fmjx-spacer>\u003Cmjx-mi size=\"s\" data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-script>\u003C\u002Fmjx-msubsup>\u003Cmjx-mi space=\"2\" data-latex=\"f\">\u003Cmjx-c class=\"mjx-c1D453\">𝑓\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"t\">\u003Cmjx-c class=\"mjx-c1D461\">𝑡\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mspace data-latex=\"\\,\" style=\"width: 0.167em;\">\u003C\u002Fmjx-mspace>\u003Cmjx-mi data-latex=\"d\">\u003Cmjx-c class=\"mjx-c1D451\">𝑑\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mi data-latex=\"t\">\u003Cmjx-c class=\"mjx-c1D461\">𝑡\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\".\">\u003Cmjx-c class=\"mjx-c2E\">.\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-tstrut>\u003C\u002Fmjx-tstrut>\u003C\u002Fmjx-mtd>\u003C\u002Fmjx-mlabeledtr>\u003C\u002Fmjx-itable>\u003C\u002Fmjx-table>\u003Cmjx-labels style=\"width: 20.029em;\">\u003Cmjx-itable align=\"right\" style=\"right: 0;\">\u003Cmjx-mtr style=\"height: 2.483em;\">\u003Cmjx-mtd id=\"mjx-eqn:eq:diffquot\">\u003Cmjx-mtext data-latex=\"\\text{(}\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mtext>\u003Cmjx-mtext data-latex=\"\\text{1}\">\u003Cmjx-c class=\"mjx-c31\">1\u003C\u002Fmjx-c>\u003C\u002Fmjx-mtext>\u003Cmjx-mtext data-latex=\"\\text{)}\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mtext>\u003Cmjx-tstrut style=\"height: 2.483em; vertical-align: -0.904em;\">\u003C\u002Fmjx-tstrut>\u003C\u002Fmjx-mtd>\u003C\u002Fmjx-mtr>\u003C\u002Fmjx-itable>\u003C\u002Fmjx-labels>\u003C\u002Fmjx-mtable>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fdiv>\n\u003Cp>By continuity of \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"f\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"f\">\u003Cmjx-c class=\"mjx-c1D453\">𝑓\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan> at \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"x\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan>, the right-hand side of \u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"\\eqref{eq:diffquot}\" class=\"NCM-N\">\u003Ca href=\"#mjx-eqn%3Aeq%3Adiffquot\">\u003Cmjx-mrow data-latex=\"\\eqref{eq:diffquot}\" class=\"MathJax_ref\">\u003Cmjx-mtext>\u003Cmjx-c class=\"mjx-c28\" noic=\"true\" style=\"padding-top: 0.748em;\">(\u003C\u002Fmjx-c>\u003Cmjx-c class=\"mjx-c31\" noic=\"true\" style=\"padding-top: 0.748em;\">1\u003C\u002Fmjx-c>\u003Cmjx-c class=\"mjx-c29\" style=\"padding-top: 0.748em;\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mtext>\u003C\u002Fmjx-mrow>\u003C\u002Fa>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container> tends to \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"f(x)\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"f\">\u003Cmjx-c class=\"mjx-c1D453\">𝑓\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"x\">\u003Cmjx-c class=\"mjx-c1D465\">𝑥\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan> as\n\u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"h \\to 0\" breakable=\"true\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"h\">\u003Cmjx-c class=\"mjx-c210E\">ℎ\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-break size=\"4\"> \u003C\u002Fmjx-break>\u003Cmjx-mo data-latex=\"\\to\">\u003Cmjx-c class=\"mjx-c2192\">→\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mn space=\"4\" data-latex=\"0\">\u003Cmjx-c class=\"mjx-c30\">0\u003C\u002Fmjx-c>\u003C\u002Fmjx-mn>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan>.\u003C\u002Fp>\n\u003Cspan class=\"env-qed\">\u003C\u002Fspan>\u003C\u002Fdiv>\n\u003C\u002Fdiv>\n\u003Ch2 id=\"a-consequence\">A consequence\u003Ca class=\"heading-anchor\" href=\"#a-consequence\" aria-hidden=\"true\" tabindex=\"-1\">#\u003C\u002Fa>\u003C\u002Fh2>\n\u003Cdiv class=\"env env-corollary\" id=\"env-2.1\">\n\u003Cdiv class=\"env-head\">\u003Cspan class=\"env-name\">Corollary\u003C\u002Fspan> \u003Cspan class=\"env-number\">2.1\u003C\u002Fspan>\u003C\u002Fdiv>\n\u003Cdiv class=\"env-body\">\n\u003Cp>If \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"G\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"G\">\u003Cmjx-c class=\"mjx-c1D43A\">𝐺\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan> is any antiderivative of \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"f\" class=\"NCM-N\">\u003Cmjx-mi data-latex=\"f\">\u003Cmjx-c class=\"mjx-c1D453\">𝑓\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan> then \u003Cspan class=\"math-inline\">\u003Cmjx-container class=\"MathJax\" jax=\"CHTML\" overflow=\"overflow\" style=\"font-size: 113.1%;\">\u003Cmjx-math data-latex=\"\\int_a^b f = G(b) - G(a)\" breakable=\"true\" class=\"NCM-N\">\u003Cmjx-msubsup data-latex=\"\\int_a^b\">\u003Cmjx-mo data-latex=\"\\int\">\u003Cmjx-c class=\"mjx-c222B\">∫\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-script style=\"vertical-align: -0.341em; margin-left: -0.12em;\">\u003Cmjx-mi size=\"s\" data-latex=\"b\" style=\"margin-left: 0.249em;\">\u003Cmjx-c class=\"mjx-c1D44F\">𝑏\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-spacer style=\"margin-top: 0.553em;\">\u003C\u002Fmjx-spacer>\u003Cmjx-mi size=\"s\" data-latex=\"a\">\u003Cmjx-c class=\"mjx-c1D44E\">𝑎\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003C\u002Fmjx-script>\u003C\u002Fmjx-msubsup>\u003Cmjx-mi space=\"2\" data-latex=\"f\">\u003Cmjx-c class=\"mjx-c1D453\">𝑓\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-break size=\"4\"> \u003C\u002Fmjx-break>\u003Cmjx-mo data-latex=\"=\">\u003Cmjx-c class=\"mjx-c3D\">=\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi space=\"4\" data-latex=\"G\">\u003Cmjx-c class=\"mjx-c1D43A\">𝐺\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"b\">\u003Cmjx-c class=\"mjx-c1D44F\">𝑏\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-break size=\"3\"> \u003C\u002Fmjx-break>\u003Cmjx-mo data-latex=\"-\">\u003Cmjx-c class=\"mjx-c2212\">−\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi space=\"3\" data-latex=\"G\">\u003Cmjx-c class=\"mjx-c1D43A\">𝐺\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\"(\">\u003Cmjx-c class=\"mjx-c28\">(\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003Cmjx-mi data-latex=\"a\">\u003Cmjx-c class=\"mjx-c1D44E\">𝑎\u003C\u002Fmjx-c>\u003C\u002Fmjx-mi>\u003Cmjx-mo data-latex=\")\">\u003Cmjx-c class=\"mjx-c29\">)\u003C\u002Fmjx-c>\u003C\u002Fmjx-mo>\u003C\u002Fmjx-math>\u003C\u002Fmjx-container>\u003C\u002Fspan>.\u003C\u002Fp>\n\u003C\u002Fdiv>\n\u003C\u002Fdiv>\n\u003Cp>This is what \u003Ca class=\"env-ref\" href=\"#thm:ftc\">Theorem 1.1\u003C\u002Fa> is for in practice: it turns the evaluation of an integral into the\nsearch for an antiderivative.\u003C\u002Fp>\n\u003Csection class=\"footnotes\" role=\"doc-endnotes\">\n\u003Ch2>Notes\u003C\u002Fh2>\n\u003Col>\n\u003Cli id=\"fn-1\">\u003Cp>The continuity hypothesis matters. Dropping it gives a version that holds almost\neverywhere, which is a genuinely different theorem. \u003Ca href=\"#fnref-1\" class=\"footnote-backref\" role=\"doc-backlink\" aria-label=\"Back to reference\">↑\u003C\u002Fa>\u003C\u002Fp>\n\u003C\u002Fli>\n\u003C\u002Fol>\n\u003C\u002Fsection>\n",[19,23],{"level":20,"text":21,"id":22},2,"The statement","the-statement",{"level":20,"text":24,"id":25},"A consequence","a-consequence",true,{"slug":28,"title":29,"url":30},"on-typography","On Typography","\u002Fpost\u002Fon-typography",{"slug":32,"title":33,"url":34},"rendering-math","Rendering Mathematics at Build Time","\u002Fpost\u002Frendering-math",1786249448752]