\begin{frame} \frametitle{Fundamental Theorem of Calculus} \vspace{-3ex} \begin{talign} g(x) = \int_a^x f(t)dt &&\implies&& g'(x) = f(x) \end{talign} \vspace{-1ex} \pause \scalebox{.8}{ \begin{tikzpicture}[default] \diagram[1]{-1}{4}{-0}{6.5}{1} \diagramannotatez \diagramannotatex{1,2,3} \diagramannotatey{1,2,3,3} \node[cred] at (2,6.5) {$f(x)$}; \draw[cred] plot[smooth,domain=-1:1.5,samples=20] (\x,{1}); \draw[cred] plot[smooth,domain=1.5:2.5,samples=20] (\x,{2}); \draw[cred] plot[smooth,domain=2.5:3.5,samples=20] (\x,{3}); \onslide<3->{ \draw[draw=none,fill=cred,opacity=.5] plot[ybar interval,domain=0.5:1.5,samples=2] (\x,{1}); } \onslide<4->{ \draw[draw=none,fill=cred,opacity=.5] plot[ybar interval,domain=1.5:2.5,samples=2] (\x,{2}); } \onslide<5>{ \draw[draw=none,fill=cred,opacity=.5] plot[ybar interval,domain=2.5:3.5,samples=2] (\x,{3}); } \onslide<6->{ \draw[draw=none,fill=cred,opacity=.5] plot[ybar interval,domain=2.5:3,samples=2] (\x,{3}); \draw[dashed,cblue] (3,-.5) -- node [at end,above] {$x = 3$} (3,6.5); \node at (1.75,.5) {area = $4.5$}; } \begin{scope}[xshift=.65\textwidth] \diagram[1]{-1}{4}{-0}{6.5}{1} \diagramannotatez \diagramannotatex{1,2,3} \diagramannotatey{1,2,3,4,5,6} \node[cred] at (2,6.5) {$g(x) = \int_{0.5}^x f(t)\,dt$}; \pause \node[cred,include] at (1.5,1) {}; \draw[cred] plot[smooth,domain=0.5:1.5,samples=20] (\x,{\x-.5}); \pause \node[cred,include] at (2.5,3) {}; \draw[cred] plot[smooth,domain=1.5:2.5,samples=20] (\x,{2*\x-2}); \pause \node[cred,include] at (3.5,6) {}; \draw[cred] plot[smooth,domain=2.5:3.5,samples=20] (\x,{3*\x-4.5}); \pause \node[include,cblue] at (3,4.5) {}; \draw[dashed,cblue] (3,-.5) -- (3,6.5); % node [at end,above] {$x = 3$} \end{scope} \begin{scope}[line width=2mm,cgreen,->,xshift=.5cm] \onslide<7->{ \draw[opacity=.7] (1.5,1) to[bend left=23] (9,4.3); \path[decoration={text along path,raise=2mm,text={area = integral\ \ \ \ \ \ },text align={center}},decorate] (1.5,1) to[bend left=23] (9,4.3); } \onslide<8->{ \draw[opacity=.7] (9,4.9) to[bend left=-25] (2.9,3.4); \path[decoration={text along path,raise=2mm,text={slope = derivative\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ },text align={center}},decorate] (2.9,3.4) to[bend left=25] (9,4.9); } \end{scope} \pause \pause \pause \end{tikzpicture} }\smallskip \mpause[0]{ Observe: $g'(x) = f(x)$ except where $f$ is not continuous.\vspace{-.75ex} } \mpause[1]{ \begin{alertblock}{} The slope (derivative) is the inverse of taking the area (integral). \end{alertblock} } \vspace{10cm} \end{frame}