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Unformatted text preview: University of Durham EXAMINATION PAPER date May/June 2006 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. ED01/2006 Durham University Copyright continued 2 MATH2051/01 SECTION A 1. Let f ( x ) = e x- sin x . Prove that the equation f ( x ) = 0 has a unique solution, p , in the interval (0 , . 8). Carry out two stages of the bisection method to estimate p , and give a bound on the magnitude of the error in your estimate. How many stages of the bisection method would be necessary to guarantee an error of no more than 10 6 ? 2. The roots x 1 and x 2 of the quadratic equation x 2 + 2 bx + c = 0 are given by x 1 =- b + b 2- c and x 2 =- b- b 2- c. Let b = 1 and c = 5 10 4 . Using three-digit arithmetic with rounding after every arithmetic operation, calculate x 1 and x 2 using the formulae above. How do these two values compare with that of your calculator to three significant figures? If there is a difference, suggest an alternative, mathematically equivalent formula which overcomes this difference....
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