\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}