\begin{frame} \frametitle{Derivatives and the Shape of a Graph} \begin{block}{First Derivative Test} Suppose that $c$ is a critical number of a continuous function $f$.\\ \begin{itemize} \pause \item If $f'$ changes the sign from positive to negative,\\ then $f$ has a local maximum at $c$. \pause\smallskip \item If $f'$ changes the sign from negative to positive,\\ then $f$ has a local minimum at $c$. \pause\smallskip \item If $f'$ does not change sign at $c$,\\ then $f$ has no local extremum at $c$. \end{itemize} \end{block} \pause[1] \begin{center} \scalebox{.7}{ \pause \begin{tikzpicture}[default,baseline=1cm] \diagram{-.5}{3}{-.2}{2.5}{1} \diagramannotatez \begin{scope}[ultra thick] \draw[cgreen] plot[smooth,domain=0:3,samples=30] function{2-(x-1.5)**2}; \tangent{1cm}{1cm}{2-pow(\x-1.5,2)}{0.75} \tangent{1cm}{1cm}{2-pow(\x-1.5,2)}{2.25} \end{scope} \end{tikzpicture} \pause\quad \begin{tikzpicture}[default,baseline=1cm] \diagram{-.5}{3}{-.2}{2.5}{1} \diagramannotatez \begin{scope}[ultra thick] \draw[cgreen] plot[smooth,domain=0:3,samples=30] function{(x-1.5)**2}; \tangent{1cm}{1cm}{pow(\x-1.5,2)}{0.75} \tangent{1cm}{1cm}{pow(\x-1.5,2)}{2.25} \end{scope} \end{tikzpicture} \pause\quad \begin{tikzpicture}[default,baseline=1cm] \diagram{-.5}{3}{-.2}{2.5}{1} \diagramannotatez \begin{scope}[ultra thick] \draw[cgreen] plot[smooth,domain=0.4:2.7,samples=30] function{1+(x-1.5)**3}; \tangent{1cm}{1cm}{1+pow(\x-1.5,3)}{0.75} \tangent{1cm}{1cm}{1+pow(\x-1.5,3)}{2.25} \end{scope} \end{tikzpicture} } \end{center} \end{frame}