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UCLA 2012 Winter Math 31B Midterm+Final Solutions. (S.Curran)

UCLA 2012 Winter Math 31B Midterm+Final Solutions. (S.Curran)

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MATH 31B, LECTURE 4 FINAL EXAMINATION MARCH 20, 2011 Name: Solutions UID: Signature: TA: (circle one) Huiyi Hu Brent Nelson Andrew Ruf Discussion meets: (circle one) Tuesday Thursday Instructions: The exam is closed-book, closed-notes. Calculators are not permitted. Answer each question in the space provided. If the question is in several parts, carefully label the answer to each part. Do all of your work on the examination paper; scratch paper is not permitted. If you continue a problem on the back of the page, please write “continued on back”. Each problem is worth 10 points. Problem Score 1 2 3 4 5 6 7 8 9 10 Total 1
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2 Problem 1: (Multiple Choice). Indicate your answer in the provided box. (i) Evaluate the infinite series: X n =0 1 + 2 n 4 n +1 . (a) 1 (b) 16/3 (c) 5/6 (d) 10/3 (e) 4/3 c (ii) Evaluate the infinite series: X n =1 2 n ( n + 2) . (a) 3/2 (b) 2 (c) 1/2 (d) 3/4 (e) 1 a (iii) Evaluate the infinite series: 1 - 1 3 + 1 5 - 1 7 + · · · . (a) π/ 2 (b) e - 1 (c) 1/2 (d) π/ 4 (e) 3/4 d (iv) Evaluate the limit: lim n →∞ n p n 2 + 1. (a) 1/2 (b) + (c) 0 (d) 1 (e) 2 d (v) Which diff. equation does y = 1 + 1 · x 3 3! + 4 · 1 · x 6 6! + 7 · 4 · 1 9! x 9 + 10 · 7 · 4 · 1 12! x 12 + · · · satisfy? (a) y 0 = x 2 y (b) y 00 = xy (c) y 00 + xy 0 - x 2 y = 0 (d) y 00 = y (e) None of these b Problem 2: (Multiple choice). Indicate the Maclaurin series (a-h) corresponding to f ( x ). (i) f ( x ) = 2 x 2+ x 2 . h (ii) f ( x ) = x ln(1 + x ). d (iii) f ( x ) = cos x 2 . a (iv) f ( x ) = tan - 1 x . g (v) f ( x ) = sin x cos x . e (a) X n =0 ( - 1) n x 4 n (2 n )! (b) X n =0 ( - 1) n x 2 n 2 n (c) X n =0 ( - 1) n x (2 n +1) (2 n + 1)! (d) X n =2 ( - 1) n x n n - 1 (e) X n =0 ( - 1) n 4 n · x 2 n +1 (2 n + 1)! (f) X n =0 ( - 1) n x 4 n +2 (2 n + 1)!
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