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128a_su09_final_review_prb

# 128a_su09_final_review_prb - MATH 128A SUMMER 2009 FINAL...

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MATH 128A, SUMMER 2009: FINAL REVIEW PROBLEMS (1) Find the largest n for which cos( x ) = 1 - x 2 2 + O ( x n ). Write the corresponding Taylor remainder. (2) Match each of (1, 2, 3) below with the correct value (a, b, c). 1. Absolute error for the estimate 5 2 2. Relative error for the estimate 5 2 3. Neither a. [ 5 - 2] / 2 b. [ 5 - 2] / 5 c. 5 - 2 (3) (a) Show that the bisection method will converge to a root of f ( x ) = x 2 - 2 in [1 , 2]. (b) Approximately how many iterations are needed to ensure two significant digits of accuracy? (4) (a) Find the unique fixed point of the following function: g ( x ) = x/ 2 + 5 /x 2 . (b) Show that the corresponding fixed point iteration p n +1 = g ( p n ) converges for any p 0 [2 , 3]. (c) Determine whether the convergence is linear or (at least) quadratic. (d) Does your answer to 4c change if Steffensen iteration is used? (5) (a) Find the unique root of the following function: f ( x ) = x/ 2 - 5 /x 2 . (b) Show that Newton’s method converges to this root if p 0 is sufficiently close. (c) Determine whether the convergence is linear or (at least) quadratic. (6) (a) How many roots (in C ) does p ( x ) = x 4 + x 2 + 2 have?

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