Math 132, Spring 2009 - Exam 3NAME:STUDENT ID NUMBER:This exam contains sixteen questions. The first fourteenare multiple choice questions and count for five pointseach. There is no partial credit on these questions, soread each question carefully, check your arithmeticand make sure that you have marked the answer you in-tended to mark. The last two questions, which are eachworth fifteen p oints, require written answers, and somepartial credit might be given. However, no credit will begiven for information that is not germane to the problemat hand. Please make sure to write your name and stu-dent ID number on the pages that include your answersto the last two questions. In fact, you will get onepoint on each of these two questions for writingyour name and ID number legibly.11. Supposedydx= 1/yand y(0) = 1. Find y(4).(a) 0(b) ±2(c) ±3(d) ±4(e) e4(f) ln 4(g) 1/4(h) −1/422. Solve the differential equationdydx= x ex2−ywith initial condition y(0) = 0.(a) ey= ex2− 1(b) ey=12ex2+12(c) ey=12ex2(d) ey= ex2(e) e−y= ex2− 1(f) e−y=12ex2+12(g) e−y=12ex2(h) e−y= ex233. Let an=(−1)nn. The sequence {an}(a) converges to −1(b) converges to 0(c) converges to 1(d) converges because it is an alternating series(e) diverges because it is an alternating series(f) diverges to ∞(g) diverges to −∞(h) diverges but not to ∞ or −∞44. Let an= (n+3n)n. Compute limn→∞an.(a) 0(b) 1(c) e(d) 3(e) π(f) e3(g) π3(h) ∞55. Compute the sum of the series∞Xn=274n(a) 0(b) ∞(c) 7/12(d) 28/3(e) 7/3(f) 21/4(g) 21/16(h) 766. Compute the sum of the series∞Xn=13n−1− 16n−1(a) 0(b) ∞(c) −∞(d) 1/3(e) 1/2(f) 1/6(g) −1/3(h) 4/577. Compute the sum of the series∞Xn=1(cos(1n) − cos(1n + 1))(a) diverges to ∞(b) diverges to −∞(c) diverges but not to −∞ or ∞(d) −1(e) −1 + cos 1(f) 0(g) 1(h) cos 188. Which of the following three series is convergent?(A)∞Xn=1n + 2n3/2(B)∞Xn=13n4− 2n2(C)∞Xn=1n√n4+ 5(a) A only(b) B only(c) C only(d) A and B only(e) A and C only(f) B and C only(g) all(h) none99. Apply the Ratio Test to∞Xn=12nn!Find ρ and, if possible, decide whether the seriesconverges or diverges.(a) ρ = 0 and the series converges by the Ratio Test.(b) ρ = 0 and the series diverges by the Ratio Test.(c) ρ = 0 and the Ratio Test fails.(d) ρ = 2 and the series converges by the Ratio Test.(e) ρ = 2 and the series diverges by the Ratio Test.(f) ρ = 2 and the Ratio Test fails.(g) ρ = ∞ and the series diverges by the Ratio Test.(h) ρ = ∞ and the Ratio Test fails.1010. Apply the Root Test to∞Xn=1(1 − 2/n)nFind ρ and, if possible, decide whether the seriesconverges or diverges.(a) ρ = e−2and the series converges by the RootTest.(b) ρ = e−2and the series diverges by the Root Test.(c) ρ = e−2and the Root Test fails.(d) ρ = 1 and the series converges by the Root Test.(e) ρ = 1 and the series diverges by the Root Test.(f) ρ = 1 and the Ro ot Te st fails.(g) ρ = ∞ and the series diverges by the Root Test.(h) ρ = ∞ and the Ro ot Test fails.1111. Which of the following three alternating series isconvergent?(A)∞Xn=1(−1)nn + 1n + 2(B)∞Xn=1(−1)n1√n(C)∞Xn=1(−1)nln nn(a) A only(b) B only(c) C only(d) A and B only(e) A and C only(f) B and C only(g) all(h) none1212. Which of the following three alternating series isabsolutely convergent?(A)∞Xn=1(−1)nn + 1n + 2(B)∞Xn=1(−1)n1√n(C)∞Xn=1(−1)nln nn(a) A only(b) B only(c) C only(d) A and B only(e) A and C only(f) B and C only(g) all(h) none1313. Find the radius of convergence for the series∞Xn=13nxnn + 2(a) R = 0(b) R = 1(c) R = 1/2(d) R = 3(e) R = 3/2(f) R = −1(g) R = x(h) R = ∞1414. For what values of x does the series∞Xn=1(x − 2)n10nconverge absolutely?(a) −1 < x < 1(b) −1 < x < 3(c) 0 < x < 3(d) 0 < x < 10(e) −8 < x < 12(f) 2 < x < 12(g) −12 < x < 12(h) −10 < x < 1015Name: Student ID:15. The series11 + t= 1 − t + t2− t3+ ... + (−t)n+ ...converges on the open interval −1 < t < 1. Find thepower series for ln(1 + x) and compute its radius ofconvergence. Show your work.16Name: Student ID:16. Use the Integral Test to decide whether∞Xn=24n ln(n)converges or diverges. Show your work. Rememberthat you must show clearly that all conditions of theIntegral Test are satisfied.17Name: Student ID:18Name: Student
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