\begin{frame} \frametitle{Derivative as a Function} Which of these functions is the derivative of the other? \medskip \begin{center} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-2.5}{2.5}{-2.5}{2.5}{1} \diagramannotatez \diagramannotatex{-2,-1,1,2} \diagramannotatey{-2,-1,1,2} \begin{scope}[ultra thick] \draw[cgreen,ultra thick] plot[smooth,domain=-1.6:1.6,samples=20] function{x**3 - x}; \end{scope} \end{tikzpicture}~\quad~% \begin{tikzpicture}[default,baseline=1cm] \diagram{-2.5}{2.5}{-2.5}{2.5}{1} \diagramannotatez \diagramannotatex{-2,-1,1,2} \diagramannotatey{-2,-1,1,2} \begin{scope}[ultra thick] \draw[cred,ultra thick] plot[smooth,domain=-1.3:1.3,samples=20] function{2*x**2 - 1}; \end{scope} \end{tikzpicture} } \end{center} \pause\medskip The right is the derivative of the left: \begin{itemize} \pause \item look at local maxima and minima of $f$; then $f'$ must be $0$ \pause \item where $f$ increases, $f'$ must be positive \pause \item where $f$ decreases, $f'$ must be negative \end{itemize} \end{frame}