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Unformatted text preview: CS 257 Numerical Methods  Homework 7 November 2, 2006 1. [1pt] 5.2 #7 Solution: We need to find the smallest integer N such that the following relationship holds 1 12 (  0) h 2 sin 00 ( ) 10 12 We can bound the left side as follows 1 12 h 2 sin 00 ( ) 1 12 h 2 sin 00 ( ) Using the fact that sin 00 ( x ) 1 and h = /N we can write 3 12 N 2 10 12 which implies r 3 10 12 12 1607437 . 8339534581 N Therefore we can chose N = 1607438 No matter how many intervals we use to compute R sin( x ) with the trapezoid rule, we will always compute something less than the true value. This is because sin is a strictly concave function on the interval [0 , ] 2. [1pt] 5.2 #12 Solution: Like the previous problem, we can bound sin 00 ( x ) 1 for x [2 , 5]. 1 12 (5 2) h 2 sin 00 ( ) 3 . 01 2 12 = 0 . 000025 3. [1pt] 6.1 #1 Solution: The approximate value is Z 1 1 1 + x 2 1 6 ( f (0) + 4 * f (0 . 5) + f (1)) = 1 6 1 + 4 4 5 + 1 2...
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This note was uploaded on 09/15/2008 for the course CS 257 taught by Professor Thomaskerkhoven during the Fall '05 term at University of Illinois at Urbana–Champaign.
 Fall '05
 ThomasKerkhoven

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