\begin{frame} \frametitle{Average Value of a Function} How to compute the \textcolor{cblue}{average value} of a function? \begin{center} \scalebox{.9}{ \begin{tikzpicture}[default] \def\mfun{(-.9 + (\x-3+\mfunshift)^2 - .1*(\x-3+\mfunshift)^4)} \diagram[1]{-.5}{6}{-1}{1.7}{1} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=.5:5.5,samples=100] (\x,{\mfun}); \node[anchor=north] at (.5,0) {$a$}; \node[anchor=north] at (5.5,0) {$b$}; \only<1>{ \draw[cblue] (.5,2.01042/5) -- (5.5,2.01042/5); } \only<2->{ \def\mwidth{5} \foreach \nrsteps/\mcolor in {6/cred} { \def\mstep{\mwidth/(\nrsteps+1)} \def\mfunshift{.5*\mstep} \foreach \xx in {0,...,\nrsteps} { \def\x{.5+ \xx*\mstep} \pgfmathparse{{\mfun}} \ifthenelse{\lengthtest{\pgfmathresult cm > 0cm}}{ \def\mcolor{cgreen} }{} \draw[thick,draw=\mcolor!60!black,fill=\mcolor,opacity=.5] ({\x},0) rectangle ({\x+\mstep},{\mfun}); \node[include=\mcolor] at ({\x+\mfunshift},{\mfun}) {}; } } } \end{scope} \end{tikzpicture} } \end{center}\vspace{-.75ex} \pause Idea: split in $n$ rectangles, take their average height. \pause \begin{talign} \frac{f(x_1) + f(x_2) + \ldots + f(x_n)}{n} \mpause[1]{ &= \frac{1}{n}\sum_{i = 1}^n f(x_i) } \mpause{= \frac{1}{n\;\Delta x}\sum_{i = 1}^n f(x_i)\Delta x} \\[-.5ex] \mpause{&= \frac{1}{b-a}\sum_{i = 1}^n f(x_i)\Delta x} \end{talign}\vspace{-1ex} \pause\pause\pause\pause \begin{block}{} The sum \quad \alert{$\sum_{i = 1}^n f(x_i) \Delta x$} \quad is called \emph{Riemann sum}. \end{block} \vspace{10cm} \end{frame}