$(3$pt$)$ Implement the Newton divided difference algorithm $($Page $332$ in the textbook, section

$6\mathrm{.}2$ with subtitle "Algorithm for Divided Difference"$)$ together with the nested multiplication

algorithm $($Page $310$ in the textbook$).$ Then use your code to find the Newton interpolating

polynomial $p_n$ for the function $f(x)=\frac{1}{1+x^2}$ on the interval $-5,5$ for $n=5,10$ and $15.$

In each case, use a smaller spacing $($e$.$g$.x=0\mathrm{.}1$ to draw the graph for $f(x)-p_n(x)$ with

xin$[-5,5].$ What do you observe?