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

    Can we at least find the root approximately?