Consider the function $f(x)=e^{-x^2}$

$($a$)$ The forward finite$-$difference formula for $f^{\text{'}}(x)$ is as follows:

$f^{\text{'}}(x)=\frac{f(x+h)-f(x)}{h}+O(h)$

Use the previous formula to estimate $f^{\text{'}}(1)$ for a step size of $h=0\mathrm{.}20.$ Calculate the $\%$ relative

error, $lon,$ as $lon=|$ exact value $\}-\{$approximate value$\}|100\%.$

$($b$)$ The centered finite$-$difference formula for $f^{\text{'}}(x)$ is as follows:

$f^{\text{'}}(x)=\frac{f(x+h)-f(x-h)}{2h}+O(h^2)$

Use the previous formula to estimate $f^{\text{'}}(1)$ for a step size of $h=0\mathrm{.}20.$ Calculate the $\%$ relative

error, $lon,$ as $lon=|$ exact value