#1 Consider the data in the following table: k 0 1 2 3 Tk 0.0 1.761062 3.522123 5.283185 Yk 1.0 -0.1891196 -0.9284676 0.5403023 The values of y in this table have been obtained as yk = cos(x). Your goal will be to create two different spline interpolants using these data points, and compare them with the original function. (a) Compute the usual spline interpolant 'S(x) that uses the natural boundary conditions. Plot this interpolant versus the original function cos(x) using 100 data points equally spaced between 20 and 23; make sure to also indicate the four data points on the plot using a special marker (you may use or o for example). (b) You will probably agree that the comparison at point (a) above doesn't look very good. This is due to the fact that the natural boundary conditions do not match the behavior of the cos(x) function. Your task for this point is to modify your code to account for the correct second derivatives at the endpoints. In other words, create a new interpolant 2S(x) that satisfies the following conditions: 2S" (xo) = cos(xo); 2S" (x3)= cos(x3). You can do this by modifying the system of equations for the spline coefficients ac- cordingly. Plot again the newly-obtained interpolant versus the original function in the same manner as above. Turn in your codes together with the two plots.