HW9solution - Homework 9 Solution 1. Given the following...

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Unformatted text preview: Homework 9 Solution 1. Given the following function: f ( x ) = 3 x 4- 4 x 3 + 2 x- 5 (a) [1.5pt] find the estimate of the first and second derivatives using the forward finite divided difference (FFDD) formulas by truncating after the first term at x = 0 . 5 and h = 0 . 25. (b) [1.5pt] repeat (a) using the FFDD formulas by keeping the first two terms. (c) [1pt] find an analytical solution and compute t for (a) and (b) Solution: (a) Truncate after the first term f ( x = 0 . 5) = f (0 . 75)- f (0 . 5) . 25 = 0 . 2969 (1) f 00 ( x = 0 . 5) = f (1)- 2 f (0 . 75) + f (0 . 5) . 25 2 = 2 . 625 (2) (b) keeping the first two terms f ( x = 0 . 5) =- f (1) + 4 f (0 . 75)- 3 f (0 . 5) 2 . 25 =- . 0313 (3) f 00 ( x = 0 . 5) =- f (1 . 25) + 4 f (1)- 5 f (0 . 75) + 2 f (0 . 5) . 25 2 =- 7 . 1250 (4) (c) exact solution f ( x ) = 12 x 3- 12 x 2 + 2 , f (0 . 5) = 0 . 5 (5) t ( a ) = | . 5- . 2969 | . 5 = 40 . 6% , t ( b ) = | . 5 + 0 . 0313 | . 5 = 106% (6) f 00 ( x ) = 36 x 2- 24 x, f 00 (0 . 5) =- 3...
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HW9solution - Homework 9 Solution 1. Given the following...

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