\begin{frame}
  \frametitle{Newton's Method}
  
  \begin{center}
  \scalebox{.8}{
  \begin{tikzpicture}[default,baseline=1cm]
    \diagram{-.5}{4}{-.5}{4}{1}
    \diagramannotatez
    \begin{scope}[ultra thick]
      \draw[cgreen,ultra thick] plot[smooth,domain=-0:3.3,samples=200] function{-.5 + (.5*x)**3};
      \node[include=cgreen] (r) at (1.58,0) {};
      \node[anchor=south east] at (r) {$r$};
    \end{scope}
    \mpause[1]{
      \draw[gray] (3.2,-.2) -- node[at start,below,black] {$x_1$} node[include=cred,at end] {} (3.2,{-.5 + (.5*3.2)^3});
    }
    \mpause{
      \tangent{4cm}{.5cm}{-.5 + (.5*\x)^3}{3.2}
    }
    \mpause{
      \draw[gray] (2.28,-.2) -- node[at start,below,xshift=1mm,black] {$x_2$} node[include=cred,at end] {} (2.28,{-.5 + (.5*2.28)^3});
    }
    \mpause{
      \tangent{1.5cm}{3.3cm}{-.5 + (.5*\x)^3}{2.28}
    }
    \mpause{
      \draw[gray] (1.77,-.2) -- node[at start,below,xshift=1mm,black] {$x_3$} node[include=cred,at end] {} (1.77,{-.5 + (.5*1.77)^3});
    }
  \end{tikzpicture}
  }
  \end{center}

  We want to find an approximation of the root $r$ of $f(x)$.
  \begin{itemize}
  \pause
    \item Take an approximation \alert{$x_1$} of the root (a rough guess).
  \pause
    \item Compute the tangent \alert{$L_1$} at $(x_1,f(x_1))$.
  \pause
    \item Find the $x$-intercept \alert{$x_2$} of the tangent $L_1$.
  \pause
    \item Compute the tangent \alert{$L_2$} at $(x_2,f(x_2))$.
  \pause
    \item Find the $x$-intercept \alert{$x_3$} of the tangent $L_2$.
  \pause
    \item \ldots continue until approximation is good enough
  \end{itemize}
  \vspace{10cm}
\end{frame}