Q#$1(50$ points$)$: Divided Differences

The set of the following five data points is given:

$\backslash$table$[[x,1,2,4,5,7],[y,52,5,-5,-40,10]]$

$($a$)$ Determine the fourth$-$order polynomial in Newton's form that passes through the points. Calculate the coefficients by using a divided difference table.

$($b$)$ Use the polynomial obtained in part $($a$)$ to determine the interpolated value for $x=3.$

$($c$)$ Write a MATLAB user$-$defined function that interpolates using Newton's polynomial. The input to the function should be the coordinates of the given data points and the $x$ coordinate of the point at which $y$ is to be interpolated. The output from the function is the $y$ value of the interpolated point.