HW2S - Homework 2 Solution 1 For the following function f(x...

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Homework 2 Solution 1. For the following function: f ( x ) = 6 x 2 + sin x (a) [1pt] Using the first forward finite divided difference method, compute the approximation of the first derivative of the above function at x i = 1 where the step size h = x i +1 - x i = 0 . 1. Repeat the process for h = 0 . 05. (b) [1pt] Find the true value of f 0 (1) analytically. (c) [1pt] Compute the true fractional relative errors for both cases (when h = 0 . 1 and h = 0 . 05) by using the analytical solution from (b). (d) [1pt] Using the second forward finite divided difference method, compute the approx- imation of the second derivative of the above function at x i = 1 where the step size h = x i +1 - x i = 0 . 1. Repeat the process for h = 0 . 05. (e) [1pt] Find the true value of f 00 (1) analytically. (f) [1pt] Compute the true fractional relative errors for both cases (when h = 0 . 1 and h = 0 . 05) by using the analytical solution from (e). Solution: (a) Approximation with first forward finite divided difference method h = 0 . 1 , f 0 (1) = f (1 . 1) - f (1) 0 . 1 = (8 . 1512 - 6 . 8415) / 0 . 1 = 13 . 0974 h = 0 . 05 , f 0 (1) = f (1 . 05) - f (1) 0 . 05 = 12 . 8190 (b) True value of f 0 (1) f 0 ( x ) = 12 x
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