$4.(20$ Points$)$ The following table gives values of $\backslash ($ f$($x$)=2\backslash$cos $\backslash$left$(\backslash$frac$\{\backslash$pi$\}\{2\}$ x$\backslash$right$)+\backslash$cos $(\backslash$pi x$)\backslash )$ for $14$ values of $\backslash ($ x $\backslash ).$ Determine the derivative, $\backslash ($ f$^\{\backslash$prime$\}($x$)\backslash ),$ exactly for those values of $\backslash ($ x $\backslash ).$ Also determine the derivative using forward, backward and centered finite differences. Determine the relative error for each of the finite difference approximations by subtracting the analytical value from your finite difference estimate. Fill in the table. Feel free to use EXCEL to make these calculations but if you want partial credit make sure to show enough work to explain your process. I don't want your excel sheet. Plot the analytical solution on the plot given below to help you check your finite difference answers.

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline x & f$($x$)$ & $\backslash (\backslash$mathrm$\{$df$\}/\backslash$mathrm$\{$dx$\}\backslash )$ & Forward & $\backslash$begin$\{$tabular$\}\{$l$\}$

Forward $\backslash \backslash$

Error

$\backslash$end$\{$tabular$\}$ & Backward & Backward Error & Center & $\backslash$begin$\{$tabular$\}\{$l$\}$

Center $\backslash \backslash$

Error

$\backslash$end$\{$tabular$\}\backslash \backslash$

$\backslash$hline $0$ & $3$ & & & & & & & $\backslash \backslash$

$\backslash$hline $0\mathrm{.}2$ & $2\mathrm{.}6$ & & & & & & & $\backslash \backslash$

$\backslash$hline $0\mathrm{.}4$ & $1\mathrm{.}7$ & & & & & & & $\backslash \backslash$

$\backslash$hline $0\mathrm{.}7$ & $0\mathrm{.}5$ & & & & & & & $\backslash \backslash$

$\backslash$hline $0\mathrm{.}9$ & $-0\mathrm{.}6$ & & & & & & & $\backslash \backslash$

$\backslash$hline $1\mathrm{.}1$ & $\backslash (-1\mathrm{.}3\backslash )$ & & & & & & & $\backslash \backslash$

$\backslash$hline $1\mathrm{.}3$ & $\backslash (-1\mathrm{.}5\backslash )$ & & & & & & & $\backslash \backslash$

$\backslash$hline $1\mathrm{.}6$ & $-1\mathrm{.}3$ & & & & & & & $\backslash \backslash$

$\backslash$hline $1\mathrm{.}8$ & $\backslash (-1\mathrm{.}1\backslash )$ & & & & & & & $\backslash \backslash$

$\backslash$hline $2$ & $-1$ & & & & & & & $\backslash \backslash$

$\backslash$hline $2\mathrm{.}2$ & $\backslash (-1\mathrm{.}1\backslash )$ & & & & & & & $\backslash \backslash$

$\backslash$hline $2\mathrm{.}5$ & $-1\mathrm{.}4$ & & & & & & & $\backslash \backslash$

$\backslash$hline $2\mathrm{.}7$ & $\backslash (-1\mathrm{.}5\backslash )$ & & & & & & & $\backslash \backslash$

$\backslash$hline $2\mathrm{.}9$ & $-1\mathrm{.}2$ & & & & & & & $\backslash \backslash$

$\backslash$hline $3\mathrm{.}1$ & $-0\mathrm{.}5$ & & & & & & & $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$