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MATH2051_05

# MATH2051_05 - University of Durham EXAMINATION PAPER date...

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University of Durham EXAMINATION PAPER date May/June 2005 exam code MATH2051/01 description NUMERICAL ANALYSIS II Time allowed: 3 hours Examination material provided: None Instructions: Credit will be given for the best FOUR answers from Section A and the best THREE answers from Section B. Questions in Section B carry TWICE as many marks as those in Section A. Approved electronic calculators may be used. ED01/2005 University of Durham Copyright continued

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2 MATH2051/01 SECTION A 1. (a) Let f ( x ) = e x - 4 x . Using the Intermediate Value Theorem prove that the equation f ( x ) = 0 has a unique solution, p , in the interval (0 , 1). (b) One rearrangement of this equation is x = g ( x ) := 0 . 25 e x . Show that the iterative method p n +1 = g ( p n ) will converge to p for suitable starting values. (c) Starting with p 0 = 0 . 36, estimate p correct to three decimal places. 2. Let f ( x ) be a smooth function defined on the interval [ - 1 , 1]. The interpolating Hermite polynomial, p 3 ( x ), of degree 3 is defined by p 3 ( - 1) = f ( - 1) , p 3 (1) = f (1) , p 0 3 ( - 1) = f 0 ( - 1) , p 0 3 (1) = f 0 (1) and satisfies the following error estimate f ( x ) - p 3 ( x ) = f (4) ( c ) 4! ( x + 1) 2 ( x - 1) 2 , c ( - 1 , 1) .
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