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\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}