review1s - 821 60 13 . 6833. Simpsons rule: 613 45 13 ....

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Numerical Analysis/Fall 2009/Review #1 Solutions to Sample Questions. (1) (a) f (1) = - 2 < 0, f (2) = 2 > 0, so by the Intermediate Value Theorem, f has a root in (1 , 2). (b) f 0 ( x ) = 3 x 2 - 3 = 3( x 2 - 1) > 0 for x (1 , 2), so f is increasing on [1 , 2], and the root is unique. (c) We solve 2 - 1 2 n < 10 - 5 to get n > log 10 5 log 2 16 . 6, so n 17. (d) Estimates are a b m 1 2 1.5 1.5 2 1.75 1.5 1.75 1.625 1.625 1.75 1.6875 1.6875 1.75 1.71875 The final estimate is 1 . 71875, and the error is at most 2 - 5 = 0 . 03125. (2) The iterating function is given by g ( x ) = x - f ( x ) f 0 ( x ) = x - tan x sec 2 x = 1 - cos x sin x. Estimates are 0 . 8, 0 . 785613, 0 . 785398, 0 . 785398. (3) f 0 ( x ) = sin x + x cos x , f 0 (0) = 0, f 0 ( π ) = - π 6 = 0, so p 0 = 0 is a multiple root and Newton’s method will con- verge linearly. p 1 = π is a simple root and Newton’s method will converge quadratically. (4) L 0 ( x ) = 1 2 ( x - 1)( x - 2), L 1 ( x ) = - x ( x - 2), L 2 ( x ) = 1 2 x ( x - 1), L ( x ) = 3 x 2 - 3 x + 1. (5) Since f (0) = 0, we need not bother finding L 0 . Then L 1 ( x ) = - 1 2 x ( x - 3), L 2 ( x ) = 1 6 x ( x - 1), L ( x ) = 2 x 2 + x , f (2) L (2) = 10. (6) Trapezoidal rule:
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Unformatted text preview: 821 60 13 . 6833. Simpsons rule: 613 45 13 . 6222. Exact: 12 + ln 5 13 . 6094. (7) (a) h 4 180 f (4) ( c ) < . 25 4 180 2 . 17 10-5 . (b) n > 10 6 180 1 / 4 8 . 63. Since n must be even, n = 10 is the minimum value that will work. (8) x f ( x ) f ( x ) 1 2 1 2 4 1 . 5 3 5 1 . 5 4 7 2 . 5 (9) 116 27 190 27 193 27 121 27 44 9 8 74 9 47 9 17 3 9 28 3 19 3 8 9 10 9 = . 000 4 . 296 7 . 037 7 . 148 4 . 481 0 . 000 . 000 4 . 889 8 . 000 8 . 222 5 . 222 0 . 000 . 000 5 . 667 9 . 000 9 . 333 6 . 333 0 . 000 . 000 8 . 000 9 . 000 10 . 000 9 . 000 0 . 000...
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This note was uploaded on 10/10/2010 for the course MATH 01:640:111 taught by Professor Carey during the Spring '10 term at Saint Louis.

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review1s - 821 60 13 . 6833. Simpsons rule: 613 45 13 ....

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