Consider

$f(x)=e^{x+1}$

$($i$)5$ point Use the Lagrange interpolation method on the nodes $x_0=0,x_1=0\mathrm{.}5,$ and $x_2=1($with $N=2),$ to find the approximation of $f(0\mathrm{.}7).$

Use the Newton interpolation method on the nodes $x_0=0,x_1=0\mathrm{.}5,$ and $x_2=1($with $N=2),$ to find the approximation of $f(0\mathrm{.}7).$

$($ii$)5$ point Compare the absolute error $|f(0\mathrm{.}7)-f_{\text{L}\text{a}\text{g}\text{r}\text{a}\text{n}\text{g}\text{i}\text{a}\text{n}}(0\mathrm{.}7)|$ and $|f(0\mathrm{.}7)-f_{\text{N}\text{e}\text{w}\text{t}\text{o}\text{n}}(0\mathrm{.}7)|,$ where $f_{\text{L}\text{a}\text{g}\text{r}\text{a}\text{n}\text{g}\text{i}\text{a}\text{n}}(0\mathrm{.}7)$ and $f_{\text{N}\text{e}\text{w}\text{t}\text{o}\text{n}}(0\mathrm{.}7)$ are approximated values obtained by Lagrangian and Newton methods, respectively.

$($iii$)5$ point Construct the general MATLAB code of the Lagrange interpolation method and the Newton interpolation method.

Note that your code has to be independent of the node size $N.$ That is$,$ you need to construct the code, which can be applied to any node size $N.$