Midterm2007_Solutions

Midterm2007_Solutions - UCLA MATH 151A/2, WINTER 2007,...

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Unformatted text preview: UCLA MATH 151A/2, WINTER 2007, MIDTERM EXAM NAME STUDENT ID # This is a closed-book and closed-note examination. No calculators are allowed. Please show all your work. Partial credit will be given to partial answers. There are 5 problems of total 100 points. You do not have to completely carry out the algebraic calculations. PROBLEM 1 2 3 4 5 TOTAL SCORE I. Let f ( x ) = 3 x e x , and the table x 1 1.125 1.250 1.375 1.500 1.625 1.750 1.875 2 f ( x ) 0.2817 0.2948 0.2597 0.1699 0.0183-0.2034-0.5046-0.8958-1.3891 (a) Prove that the equation f ( x ) = 0 has at least a solution p in the interval [1 , 2]. Solution: From the table, we see that f (1) = 0 . 2817 > 0, while f (2) = 1 . 3891. Also, f is continuous on [1 , 2], thus by the Intermediate Value Thm., there must be a p (1 , 2) such that f ( p ) = 0. (b) By the Bisection method, find p n , n 2 on [1 , 2], and write your answers in the next table. n a n b n p n f ( p n ) 1 2 1.5 0.0183 1 1.5 2 1.75-0.5046 2 1.5 1.75 1.625-0.2034 Solution: (c) How many iterations are necessary to solve 3 x e x = 0 with accuracy 10 4 on [1 , 2] ? Solution: From the theorem from the course, we impose | p n p | b a 2 n 10 4 , thus we impose 2 1 2 n 10 4 , or 10...
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This note was uploaded on 10/28/2010 for the course MATH 151a taught by Professor Staff during the Spring '08 term at UCLA.

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Midterm2007_Solutions - UCLA MATH 151A/2, WINTER 2007,...

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