Do in Python. Compute the second derivative $d2\frac{u}{d}x2$ of the function $u=sin(x)$ in the domain Doxin$[0,2]$ for $h=0\mathrm{.}1,0\mathrm{.}05,0\mathrm{.}025$ and

$0\mathrm{.}01$ using the centered difference scheme:

$\frac{d^{2}u}{d{x}^2}$~~$\frac{u_{j-1}-2u_j+u_{j+1}}{h^2},$

Plot both the discretized derivatives and the analytic derivative on a single graph. Do not forget to label the axes and the

legend explaining what each line is$.$

Then, compare the results with the exact solution by calculating the maximum error $($i$.$e$.,$ the maximum of the absolute

error$)$ and answer How does the maximum error change with the grid spacing $h?$ Plot the maximum error $(y-$axis$)$ as a

function of the grid spacing $h(x-$axis$).$