Use the three$-$point centered$-$difference formula for second derivative

$f^{**}(x)=\frac{1}{h^2}(f(x+h)-2f(x)+f(x-h))$

a$)$ to approximate $f^{\text{'}\text{'}}(x)$ at $x=1,$ where $f(x)=arctan(x),$ for $h={10}^n,n=-1,-2,$dots,$-17$;

b$)$ prove that the error of the approximation, i$.$e$.,f^{\text{'}\text{'}}(x)-f^{\text{'}\text{'}}(x)-O(h^2).$

Use the three$-$point centered$-$difference formula for second derivative

$f^{**}(x)=\frac{1}{h^2}(f(x+h)-2f(x)+f(x-h))$

a$)$ to approximate $f^{\text{'}}(x)$ at $x=1,$ where $f(x)=arctan(x),$ for $h={10}^n,n=-1,-2,$dots,$-17$;

b$)$ prove that the error of the approximation, i$.$e$.,f^{\text{'}\text{'}}(x)-f^{\text{'}\text{'}\text{'}}(x)-O(h^2).$

Note: The results of a$)$ should be presented in the form of a table, an example of which is given below:

$\backslash$table$[[h,($:$f_{\text{M}\text{i}\text{m}}(1)$