Math 128A, Spring 2016 Problem Set 05 Question 1 Let p be a positive integer and f (x) = 2x for 0 x 2. (a) Find a formula for the pth derivative f (p) (x). (b) For p = 0, 1, 2 nd a formula for the polynomial Hp of degree 2p + 1 such that (k) Hp (xj ) = f (k) (xj ) for 0 k p, 0 j 1, x0 = 0, x1 = 2. (c) For general p prove that |f (x) Hp (x)| 2p+2 1 p+1 for 0 x 2. (d) Show that one step of Newton's method for solving g(y) = x ln 2 ln y = 0 starting from y0 = H6 (x) gives y1 = f (x) = 2x to full double precision accuracy for 0 x 2. Question 2 (See BF p. 192.) For integer k 4 let pk = k sin Pk = k tan k k (a) Show that p4 = 2 2 and P4 = 4. (b) Show that P2k = 2pk Pk pk + P k p2k = pk P2k for k 4. (c) Approximate within 104 by computing pk and Pk until Pk pk < 104 . (d) Use Taylor series to show that = pk + qj k 2j Qj k 2j = Pk + j=1 j=1 for some constants qj and Qj . (e) Use extrapolation with h = 1/k to approximate within 1012 . 1