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Unformatted text preview: Math 471 Midterm 1 17 October 2007, 68 pm Name: Instructor: • Show all work and circle your final answers. • If you need additional space, continue on the back of the page or on the extra sheet at the end of the exam. • No calculators allowed. Problem Possible Points Score 1 20 2 15 3 15 4 10 5 20 6 20 Total 100 1 1. (True/False) Justify your answer for full credit. (a) [5 pts] Suppose the value of a function f at certain points x is given by x f ( x ) 1.2 3.64 0.6 3.16 0.3 3.04 0.15 3.01 The data suggest that f is converging to 3 as x → 0 with rate of convergence O( x ). False: lim x →  f ( x ) 3  ≈ 4 9 x 2 . So the rate of convergence should be O ( x 2 ). (b) [5 pts] Set f ( x ) = cos( x ) 1 and note f (2 π ) = 0. Using g ( x ) = x 2 2! + x 4 4! in place of f ( x ) is an effective way to reduce the error from cancellation for x close to 2 π . False: g ( x ) is the first two terms of the Taylor expansion of f ( x ) near 0, NOT 2 π . So if we want to reduce the error near 2 π , we should replace x in g ( x ) with ( x 2 π ). 2 (c) [5 pts] Using three digit rounding decimal arithmetic, the value of 2 × 10 2 · (4 × 10 3 3 × 10 ) + 1 × 10 3 is 8 . 01 × 10 5 . True (Details skipped here but you should write them out) (d) [5 pts] Newton’s method will perform better finding the unique root of f ( x ) = x 3 2 than in finding the unique root of g ( x ) = ( x 2) 3 ....
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 Winter '08
 LinZhi
 Numerical Analysis, Addition, pts, matrix norm

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