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Introduction to Numerical AnalysisMath 104A, Winter 2009Instructor:Carlos J. Garc´ıa-CerveraDecember 8th, 2010Answer the following 7 questions. Show all your work for full credit. You mustinclude your computer programs. Follow the guidelines for presentation of results.Name:Due on Wednesday December 8th, before 12pm.Problem 1:out of 10.Problem 2:out of 10.Problem 3:out of 25.Problem 4:out of 20.Problem 5 (BONUS):out of 20.Problem 6:out of 35.Problem 7:out of 40.Total:out of 140.THESE SHEETS ARE TO BE HANDED IN WITH YOUR EXAM.Page 1 of 5Please go on to the next page. . .
Math 104AFinal ExamFall 20101. (10 points) Consider the following iterative method to solvef(x) = 0:xn+1=g(xn),whereg(x) =x-f(x)f(x)-12f(x)f(x)f(x)f(x)2.(a) (5 points) Does the method converge? What is the rate of convergence of the method?Justify your answer.(b) (5 points) Find, using this method, and to within 10-10, the solution toe-x= sin(x)located in the interval [0,1].2. (10 points) Find the coefficientsa0,a1,a2, anda3, such thata0f(x-2h) +a1f(x-h) +a2f(x+h) +a3f(x+ 2h) =f(x) +O(h4)for any smooth functionf. Check your answer withf(x) =excos(x) atx= 0. Use severalvalues ofhand show numerically that the degree of accuracy of the formula that you haveobtained is correct.3. (25 points) The semi perimeters of regular polygons with