\begin{frame} \frametitle{Derivatives and the Shape of a Graph} \begin{exampleblock}{Curve Sketching} \begin{malign} f(x) = x^4 - 4x^3 = x^3 (x - 4) && f'(x) &= 4x^2(x - 3) \end{malign}\vspace{-1ex} \begin{itemize} \pause \item $f(x) = 0 \;\iff\;$ \pause \textcolor{cblue}{$x = 0$} \quad or \quad \textcolor{cblue}{$x = 4$} \pause \item local minimum at \textcolor{cdgreen}{$(3,-27)$} and \textcolor{cdgreen}{$f'(0) = 0$} \pause \item inflection points \textcolor{cred}{$(0,0)$} and \textcolor{cred}{$(2,-16)$} \pause \item decreasing on \pause $(-\infty,0)$ and $(0,3)$, \pause increasing on \pause $(3,\infty)$ \pause \item concave up on $(-\infty,0)$, down on $(0,2)$, up on $(2,\infty)$ \end{itemize} \pause[1] \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-2.5}{6}{-2}{2}{1} \diagramannotatez \diagramannotatex{-2,-1,1,2,3,4,5} \diagramannotateyy{-1/-16,1/16} \mpause[2]{ \node[include=cblue,minimum size=4mm] at (0,0) {}; \node[include=cblue] at (4,0) {}; } \mpause[3]{ \node[include=cdgreen] at (3,-27/16) {}; \node[include=cdgreen,minimum size=3mm] at (0,0) {}; } \mpause[4]{ \node[include=cred] at (0,0) {}; \node[include=cred] at (2,-16/16) {}; } \begin{scope}[ultra thick] \mpause[10]{ \draw[cgreen] plot[smooth,domain=-1.75:0,samples=30] function{(x**4 - 4*x**3)/16}; } \mpause[11]{ \draw[cgreen] plot[smooth,domain=-0:2,samples=30] function{(x**4 - 4*x**3)/16}; } \mpause[12]{ \draw[cgreen] plot[smooth,domain=2:3,samples=30] function{(x**4 - 4*x**3)/16}; } \mpause[13]{ \draw[cgreen] plot[smooth,domain=3:4.35,samples=30] function{(x**4 - 4*x**3)/16}; } \end{scope} \end{tikzpicture} } \end{center} \end{exampleblock} \end{frame}