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

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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 0 ( x = 0 . 5) = f (0 . 75) - f (0 . 5) 0 . 25 = 0 . 2969 (1) f 00 ( x = 0 . 5) = f (1) - 2 f (0 . 75) + f (0 . 5) 0 . 25 2 = 2 . 625 (2) (b) keeping the first two terms f 0 ( x = 0 . 5) = - f (1) + 4 f (0 . 75) - 3 f (0 . 5) 2 × 0 . 25 = - 0 . 0313 (3) f 00 ( x = 0 . 5) = - f (1 . 25) + 4 f (1) - 5 f (0 . 75) + 2 f (0 . 5) 0 . 25 2 = - 7 . 1250 (4) (c) exact solution f 0 ( x ) = 12 x 3 - 12 x 2 + 2 , f 0 (0 . 5) = 0 . 5 (5) t ( a ) = | 0 . 5 - 0 . 2969 | 0 . 5 = 40 . 6% , t ( b ) = | 0 . 5 + 0 . 0313 | 0 . 5 = 106% (6) f 00 ( x ) = 36 x 2 - 24 x, f 00 (0 . 5) = - 3 (7) t ( a ) = | - 3 - 2 . 625 | 3 = 187% , t ( b ) = | - 3 + 7 . 125 | 3 = 137 . 5% (8) 2. Numerically solve the following differential equation from x = 0 to x = 0 . 5 using h = 0 . 25: dy dx = 3 x - 2 y where y | x =0 = y (0) = 2. (a) [1pt] using the Euler’s method. (b) [1pt] using the Heun’s method.
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