P$4\mathrm{.}3\mathrm{.}1\frac{f(x+h)-f(x-h)}{2h}$ is a centered derivative approximation of $f^{\text{'}}(x).$ Using an awk script,

determine the optimal choice of $h=1,{}^{\frac{1}{3}},{}^{\frac{2}{3}},$ when approximating the derivative of the

function $f(x)=x^4$ at $x=2$ with a centered derivative approximation. Hint: compute the absolute error.