$3\mathrm{.}7$ We turn to the first central$-$difference approximation to the second derivative.

$($a$)$ Start with the error analysis, including both approximation and roundoff error.

Derive expressions for the hopt and the Eopt. Then, produce numerical estimates

for hopt and the Eopt. Compare these results to those for the first derivative.

$($b$)$ Now code this problem up in Python $($for the function f$($x$)=$ e

sin$(2$x$)$

at x $=0\mathrm{.}5)$

to produce both a table of numbers and a plot for the absolute error, with h taking

on the values $101$

$,102$

$,103$

$,...,1010$