Problem $4$

Verify your fourth$\_$order$\_$diff$()$ function by performing a convergence analysis on the following function:

$f(x)=$exp$(-x)$

To do this, let's select a single point $x=0\mathrm{.}6$ and a series of heights $h=1,0\mathrm{.}5,\frac{1}{3},0\mathrm{.}25,0\mathrm{.}2,\frac{1}{6},\frac{1}{7},\frac{1}{8},\frac{1}{9}.$ For each height, compute the error between the numerical derivative and the exact derivative, $-$exp$(-0\mathrm{.}6).$ The error is the absolute value of the difference between the output of your function, and the value of the exact derivative.

In your write up$,$ include a line plot of $h$ vs error. Then, verify that we see $O(h^4)$ convergence rate by estimating the convergence rate the following way: Ratei$=$log$($errori$+1/$errori$)/$log$($hi$+1/$hi$)$ for i$=1,2....,8$

In your write up print this rate vector and include a screenshot of your code which generated the plot and estimated the convergence rate.