The objective of this problem is to explore polynomial splines through a combination of explicit calculations and some light programming. We didn't get to cover this in class and this problem is meant to fill in the gap. Recall the definition of a spline. Given n 3 1 distinct points {My-\"Ll a polynomial spline p of order if is such that o pm) my; for i E l;n]]: o p is an Eth order polynomial between consecutive points x;- and rig-+1; - p{x} has it 1 continuous derivatives at the interior points of {in L1 [that is the points x.- that are not the largest and smallest; we can't define continuity at these extreme points}: In the following, we try to interpolate between four points using splines of different orders. Assume the points where we interpolate are {E}. 1.2. 3} and the corresponding values are {3. 1.0.5.1}. We denote the points by {x.}?=1 and the corresponding values by {J'}?=1. [Q1] Find a polynomial F of degree 3 that interpolates the data. i.e.. f is of the Form ier} = 21:1 and and x.) = Jag. You can ultimately use a numerical solver [python preferred but you can use l'v'latlab}. but explain how you setup the problem you solve. [02] Plot the resulting polynomial interpolation on the interval [(1.3] and show the original points. as well. [133] Suppose we want to find an order 2 spline that interpolates the data. How many unknowns are there? How many equations restrict these unknowns? How can you tryr to make the solution unique? (Hint: think about the additional constraints that you may want to add on extreme points} [as] Plot the resulting spline interpolation of order 2 on the interval [I13] and show the original poi nts. as well. [Q5] Suppose we want to find an order 3 spline that interpolates the data. How many unknowns are there? How many equations restrict these unknowns? Add the constraints that the second derivatives at x0 and x3 are zero. [Q6] Plot the resulting spline interpolation of order 3 on the interval [I13] and show the original poi nts, as well. [Q7] What happens if we impose the constraint that the third derivative of the order 3 spline are contin uous at the interior points x; and x3? The splines of order 3 are called cubic splines. Cubic splines are often viewed as the lowest-order spline for which the knotdiscontinu'rty is not visible to the human eye. There is seldom any good reason to go beyond cubicsplines. unless one is interested in smooth derivatives