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HW1S - Homework 1 1[2pts Derive Taylor series expansion for...

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Homework 1 1. [2pts] Derive Taylor series expansion for the following function from x i = 0: f ( x ) = e x . Solution: Since f ( n ) ( x i ) = e x i = 1 for n = 1 , 2 , . . . and f ( x i ) = 1, we have f ( x ) = f ( x i ) + X n =1 f ( n ) ( x i ) n ! ( x - x i ) n = X n =0 x n n ! 2. For the following function: f ( x ) = sin x (a) [2pts] Derive zero to third order Taylor series approximation of f ( x i +1 ) about a point x i = 0 . 9 with a step size h = x i +1 - x i = 0 . 1. (b) [2pts] Compute the true value of f ( x i +1 ) directly from the given function and find true relative errors for each approximation. Discuss the results. Solution: (a) Zero order Taylor series approximation f (1) = sin 1 = sin 0 . 9 = 0 . 78333 First order Taylor series approximation f (1) = sin 1 = sin 0 . 9 + cos 0 . 9 × 0 . 1 = 0 . 84549 Second order Taylor series approximation f (1) = sin 1 = sin 0 . 9 + cos 0 . 9 × 0 . 1 - sin 0 . 9 × 0 . 1 2 2! = 0 . 84157 Third order Taylor series approximation f (1) = sin 1 = sin 0 . 9 + cos 0 . 9 × 0 . 1 - sin 0 . 9 × 0 . 1 2 2! - cos 0 . 9 × 0 . 1 3 3! = 0 . 84147 (b) The true value is f (1) = sin 1 = 0 . 84147 For zero order ε a = 0 . 84147 - 0 . 78333 0 . 84147 = 0 . 0691 For first order ε a = 0 . 84147 - 0 . 84549 0 . 84147 = - 0 . 0048 1
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