1. Let f(x) = sin 2r and P(r) be the interpolation polynomial off with degree n at zo... Show that 2n+1 \f(x) - P(x) (n+1)! (x - )(x - )(x - x). 2 Given f(x) = 1 and =1+i/20, i = 0,1,20. Note that ro = 1, 220 = 2. A. Write the formula of the piecewise polynomial S(r) on the interval [ri, +1] for i= 0, 1,, 19. B. Use the error estimate theorem to show that f(x) - S(x)| 1600 C. Verify the error estimate with = 1.42 3. Given data (2, 1), (3, 1.2) and (4, 0.4). Construct the natural spline function S(x). Matlab assignment 4. Given data ( profile of a duck) r=[0.9 1.3 1.9 2.1 2.6 3.0 3.9 4.4 4.7 5.0 6.0 7.0 8.0 9.2 10.5 11.3 11.6 12 12.6 13 13.3] and y=[1.3 1.5 1.85 2.1 2.6 2.7 2.4 2.15 2.05 2.1 2.25 2.3 2.25 1.95 1.4 0.9 0.7 0.6 0.5 0.4 0.25). Draw the data and the curves of Lagrange polynomial, piecewise linear approximation, and the cubic spline. For cubic spline, use the Matlab command spline