tutorial_wk4

tutorial_wk4 - Tutorial week 4 1. (a) Let f (x) = ex 4x....

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Tutorial week 4 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. ————————————————————————————————————- (a) Firstly noting that f (0) = 1 and f (1) = e - 4 < 0 it follows from the Intermediate Value Theorem that there is at least one root p . Secondly, since f 0 ( x ) = e x - 4 < 0 for x [0 , 1] it follows that f is monotone decreasing and so there is at most one root. Hence there is a unique solution in the interval (0 , 1). (b) It is sufficient to show that | g 0 ( p ) | < 1, since then the Contraction Mapping theorem applies in a neighbourhood of p . Noting that p [0 , 1] and that for
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This note was uploaded on 02/09/2011 for the course MATH 2051 taught by Professor Dr.a.wacher during the Fall '11 term at Durham.

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tutorial_wk4 - Tutorial week 4 1. (a) Let f (x) = ex 4x....

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