\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}