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MATH 172.504 Examination 2 March 8, 2012 NAME SIGNATURE “An Aggie does not lie, cheat, or steal or tolerate those who do.” This exam consists of 8 problems, numbered 1–8. For partial credit you must present your work clearly and understandably and justify your answers. The use of calculators is not permitted on this exam. The point value for each question is shown next to each question. CHECK THIS EXAMINATION BOOKLET BEFORE YOU START. THERE SHOULD BE 8 PROBLEMS ON 6 PAGES (INCLUDING THIS ONE). Do not mark in the box below. 1–5 6 7 8 Total Points Possible Credit 30 40 12 12 94 NAME MATH 172 Examination 2 Page 2 Multiple Choice: [6 points each] In each of Problems 1–5, circle the best answer. Z 1. π x cos(x) dx. Evaluate 0 (A) 2 (B) 1 (C) 0 (D) −1 (E) −2 2. After trigonometric substitution, the integral Z √ x4 x2 − 4 dx becomes Z (A) 16 Z (B) 32 Z (C) 64 Z (D) 32 Z (E) 64 tan4 θ sec3 θ dθ tan θ sec4 θ dθ tan2 θ sec5 θ dθ sin4 θ cos θ dθ sin4 θ cos2 θ dθ March 8, 2012 NAME MATH 172 Z 3. Evaluate 1 ∞ (x2 Examination 2 Page 3 4x dx. + 1)2 (A) 0 (B) 1 (C) 2 (D) 4 (E) The integral diverges. 4. Consider the following improper integrals: Z 1 Z 0 1 I. dx II. ex dx 2 x −1 −∞ Z III. 1 ∞ 1 √ dx x Which of these integrals converge? (A) I only (B) II only (C) III only (D) I and II only (E) I and III only March 8, 2012 NAME 5. MATH 172 Examination 2 Page 4 If we use Simpson’s rule to approximate Z 2 f (x) dx 0 with 4 equal subintervals, we obtain which expression below? 6. (A) 1 f (0) + 4f ( 12 ) + 2f (1) + 4f ( 23 ) + f (2) 6 (B) 1 f (0) + 4f ( 12 ) + 2f (1) + 4f ( 32 ) + f (2) 12 (C) 1 f (0) + 2f ( 12 ) + 4f (1) + 2f ( 23 ) + f (2) 12 (D) 1 f (0) + 2f ( 12 ) + 2f (1) + 2f ( 32 ) + f (2) 8 (E) 1 f (0) + 2f ( 12 ) + 2f (1) + 2f ( 23 ) + f (2) 2 [40 points] Evaluate the following integrals. (This problem continues on the next page.) Z (a) sin(x)e2x dx March 8, 2012 NAME MATH 172 Examination 2 Page 5 Z (b) sin(7x) sin(3x) dx Z 3x2 √ dx 1 − 9x2 Z x2 − x + 3 dx x3 + x (c) (d) March 8, 2012 NAME 7. MATH 172 Examination 2 Page 6 [12 points] Find all solutions of the differential equation y dy · − sin3 (x) = 0. 4 cos (x) dx 8. [12 points] Solve the initial value problem, x dy − y = x ln x, dx y(e) = 2e. March 8, 2012