6/50
\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}
\end{tikzpicture}
}
\end{center}
\pause

Assume we want to find a root of a complicated function like:
\begin{talign}
f(x) = x^{7} - x + \cos x
\end{talign}
\pause
Often it is impossible to solve such equations!
\pause
E.g. there are no formulas for solutions of polynomials of degree of $\ge 5$.
\pause

\begin{exampleblock}{}
Can we at least find the root approximately?
\end{exampleblock}
\vspace{10cm}
\end{frame}