\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}