\begin{frame} \frametitle{Derivatives and the Shape of a Graph} \begin{exampleblock}{} What are the local extrema of \begin{talign} f(x) &= x + 2\sin x && \text{$0 \le x \le 2\pi$ \quad ?} \end{talign}\vspace{-.5ex} \pause We have \begin{talign} &f'(x) = \mpause[1]{1 + 2\cos x} \\[-1ex] &\mpause[2]{f'(x) = 0 \;\iff\;} \mpause[3]{ \cos x = -\frac{1}{2} \;\iff\;} \mpause[4]{ x = \frac{2\pi}{3} \text{ or } x = \frac{4\pi}{3} } \end{talign} \pause\pause\pause\pause\pause As $f'$ is defined everywhere these are the only critical numbers. \begin{overlayarea}{\textwidth}{2.6cm} \only<-21>{ \begin{center}\vspace{-.5ex} \begin{tabular}{|c|c|l|} \hline Interval & $f'(x)$ & \\ \hline \mpause[1]{ $0 < x < \frac{2\pi}{3}$ } & \mpause[4]{+} & \mpause{ increasing on $(0,\frac{2\pi}{3})$ } \\ \hline \mpause[2]{ $\frac{2\pi}{3} < x < \frac{4\pi}{3}$ } & \mpause[6]{-} & \mpause{ decreasing on $(\frac{2\pi}{3},\frac{4\pi}{3})$ } \\ \hline \mpause[3]{ $\frac{4\pi}{3} < x < 2\pi$ } & \mpause[8]{+} & \mpause{ increasing on $(\frac{4\pi}{3},2\pi)$ } \\ \hline \end{tabular} \end{center} } \only<22>{ \begin{center}\vspace{-1ex} \scalebox{.9}{ \begin{tikzpicture}[default,baseline=1cm,yscale=.4] \diagram{-.5}{7}{-.2}{6}{1} \diagramannotatez \diagramannotatey{2,4} \diagramannotatexx{pi/$\pi$,2*pi/$2\pi$} \begin{scope}[ultra thick] \draw[cgreen] plot[smooth,domain=0:2*pi,samples=30] function{x + 2*sin(x)}; \end{scope} \end{tikzpicture}\vspace{-1ex} } \end{center} } \end{overlayarea} \pause[17] As a consequence: \begin{itemize} \pause \item $f(\frac{2\pi}{3}) = \frac{2\pi}{3} + \sqrt{3}$ is \pause a local maximum \textcolor{gray}{($f'$ from $+$ to $-$)} \pause \item $f(\frac{4\pi}{3}) = \frac{4\pi}{3} - \sqrt{3}$ is \pause a local minimum \textcolor{gray}{($f'$ from $-$ to $+$)} \end{itemize} \end{exampleblock} \end{frame}