Question

. Use Richardson's extrapolation method to compute the derivative of fx)- e' sinx) at x - I with initial Δx = 0.1. You should carry out the extrapolation steps until the computation for Ax-0.025 completes, and clearly indicate the desired answer. You should show all computation steps. Only providing an answer and/or using a wrong method receives zero point.

Answer 1

f(x) = e^x sin (x) f' (x) = ? at x = 1 with initial increment x = 0.1 Table for Richardson's Extropolation: - O (h^2) O (h^4) O(h^6) 1 . N_1 (h) 2. N_1 (h/2) 3. N_2(h) 4. N_1 (h/4) 5.N_2 (h/2) 6.N_3 (h) Here h = 0.1 Now, using forward difference formula, N_1 (h) = N_1 (0.1) = f(x + 0.1) - f(x)/0.1 put x = 1 n_1 (0.1) = f(1.1) - f(1)/0.1 = e^1.1 sin (1.1) - s^1 sin (1)/0.1 = 3.899795 \r\n N_1 (h/2) = N_1 (0.05) = f(1.05) - f(1)/0.05 = 3.828753 Using first stage Richardson Extrapolation: - N_2 (h) = 2N_1 (h/2) - N_1 (h) = 2(3.828753) - 3.899795 N_2 (h) = 3.757711 (1) Now, N_1 (h/4) = N_1 (0.025) = f(1.025) - f(1)/0.025 = 3.79259 Now, N_2 (h/2) = 2N_1 (h/4) - N-1 (h/2) = 2(3.79259) - 3.828753 N_2(h/2) = 3.756427 (2) \r\n Second stage Richardson Extrapolation