\begin{frame} \frametitle{Average Value of a Function} \begin{block}{} The average value $f_{\text{avg}}$of a function $f$ on an interval $[a,b]$ is: \begin{talign} f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x) \, dx \end{talign} \end{block} \pause\medskip This is \emph{easy to remember}: \begin{center} \scalebox{.9}{ \begin{tikzpicture}[default,yscale=.7] \def\mfun{(-.9 + 1.2+ (\x-3+\mfunshift)^2 - .1*(\x-3+\mfunshift)^4)} \diagram[1]{-.5}{6}{-.25}{2.75}{1} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[draw=none,fill=cblue!30] plot[smooth,domain=.5:5.5,samples=100] (\x,{\mfun}) -- (5.5,0) -- (.5,0) -- cycle; \draw[cblue] plot[smooth,domain=.5:5.5,samples=100] (\x,{\mfun}); \node[anchor=north] at (.5,0) {$a$}; \node[anchor=north] at (5.5,0) {$b$}; \mpause[3]{ \draw[draw=none,fill=cblue,opacity=.3] (.5,1.2+2.01042/5) -- (5.5,1.2+2.01042/5) -- (5.5,0) -- (.5,0) -- cycle; \draw[cblue] (.5,1.2+2.01042/5) -- (5.5,1.2+2.01042/5); } \end{scope} \end{tikzpicture} } \end{center}\vspace{-.75ex} \begin{itemize} \pause \item Think of the area below the function as water. \pause \item Then the amount of water is $A = \int_a^b f(x) dx$ \pause \item When the waves calm, the water settles in the shape of a rectangle \pause with area $A$ \pause and width $b-a$; \pause thus height $\frac{A}{b-a}$ \end{itemize} \vspace{10cm} \end{frame}