\begin{frame}
  \frametitle{Derivatives and the Shape of a Graph}

  \begin{block}{}
    Let $I$ be an interval. If the graph of $f$ is called
    \begin{itemize}
    \pause
      \item \emph{concave up} on $I$ if it it lies above all its tangents on $I$
    \pause
      \item \emph{concave down} on $I$ if it it lies below all its tangents on $I$
    \end{itemize}
  \end{block}
    \pause[1]\medskip
    
    \begin{center}\vspace{-1ex}
    \scalebox{.8}{
    \pause
    \begin{tikzpicture}[default,baseline=1cm]
      \diagram{-.5}{3.5}{-.5}{3.5}{1}
      \diagramannotatez
      \begin{scope}[ultra thick]
        \draw[cgreen] plot[smooth,domain=.5:3,samples=30] function{.5+x**3/10};
        \node[include=cgreen] (na) at (.5,.5+.5^3/10) {};
        \node[include=cgreen] (nb) at (3,.5+3^3/10) {};
      \end{scope}
      \draw[gray] (na) -- node[at end,  below,black] {$a$} (.5,-.25);
      \draw[gray] (nb) -- node[at end,below,black] {$b$} (3,-.25);
      \tangent{1cm}{1cm}{.5+pow(\x,3)/10}{1.5}
      
      \node at (1.75,-1.3) {concave up};
    \end{tikzpicture}\hspace{1cm}
    \pause
    \begin{tikzpicture}[default,baseline=1cm]
      \diagram{-.5}{3.5}{-.5}{3.5}{1}
      \diagramannotatez
      \begin{scope}[ultra thick]
        \draw[cgreen] plot[smooth,domain=.5:3,samples=30] function{3.2-(3.5-x)**3/10};
        \node[include=cgreen] (na) at (.5,.5+.5^3/10) {};
        \node[include=cgreen] (nb) at (3,.5+3^3/10) {};
      \end{scope}
      \draw[gray] (na) -- node[at end,  below,black] {$a$} (.5,-.25);
      \draw[gray] (nb) -- node[at end,below,black] {$b$} (3,-.25);
      \tangent{1cm}{1cm}{3.2-pow(3.5-\x,3)/10}{2}
      
      \node at (1.75,-1.3) {concave down};
    \end{tikzpicture}
    }
    \end{center}
    
    \mpause[1]{
    Imagine the graph as a street \& a car driving from left to right:
    \begin{itemize}
    \pause\pause
      \item then concave upward = turning left \pause (increasing slope)
    \pause
      \item then concave downward = turning right \pause (decreasing slope)
    \end{itemize}
    }
\end{frame}