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Unformatted text preview: Ma 449: Numerical Applied Mathematics. Solutions to Final Examination. Prof. Wickerhauser; 6:008:00pm Friday, December 11th, 2009 You may use a calculator and the textbook with any notes you wrote in the textbook or on an 8.5 by 11 inch sheet of paper. Please write your answers in the bluebook. 1. Suppose f 000 is continuous and bounded by M 3 . Use Taylors theorem to estimate the error in the difference formula f ( x ) [ f ( x + h/ 2)- f ( x- h/ 2)] /h , in terms of M 3 , as h 0. Solution: By Taylors theorem: [ f ( x + h/ 2)- f ( x- h/ 2)] /h = f ( x )- ( h 2 / 8)[ f 000 ( c ) + f 000 ( d )] / 6 for some c, d between x- h/ 2 and x + h/ 2. The order of approximation is therefore O ( h 2 ) with constant M 3 / 24. 2 2. Suppose that Q ( h ) = Q ( f, [ a, b ] , h ) is a quadrature rule that satisfies Q ( h ) = Z b a f ( x ) dx + O ( h 4 ) , depends smoothly on h , and is an even function of h : Q (- h ) = Q ( h ) for all h . Find a formula that combines Q ( h ) and Q ( h/ 2) to give R...
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