f(x) = 1/(1+25x?) This is a somewhat famous function in numerical analysis because it displays interesting behavior when approximated using polynomial interpolation on uniformly spaced nodes. You are going write your own code for polynomial interpolation and investigate the accuracy of two approximations. Task: Do the following: 1. Write a function that takes an array containing n +1 distinct points and returns the coefficients a, k 0, 1,...,n for the Newton form of the interpolating polynomial for an arbitrary function. You can pass in either the function f(x) or an array of function values at the points. 2. Write a function that takes the Newton coefficients of the polynomial and a point x and evaluates P(x). This should be the nested product form that is similar to Horner's method. 3. Write a program that calls your two functions to compute the interpolating polynomial for the specific function f(x) given above and does the following: (a) Using the 21 uniformly spaced nodes Xk = -1+ k(.1), k = 0, 1, ... , 20 have your program compute the error at r = .985 and also at r = 0.1. (b) Using 21 nodes given by 2k + 1 Ik = Cos k = 0, 1, ..., 20. 2(20+1) have your program compute the error at x = 0.985 and also at x = 0.1.