Math 132, Spring 2009 - Final ExamNAME: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. ComputeZdxx lnx(a) cos(ex) + C(b) ecos x+ C(c) x sin(ex) + C(d) −ln |x| + C(e) ln |x| + C(f)1ln(x)+ C(g) ln |ln x| + C(h)1x+ C22. Computedydxwheny =Zx20cos√t dt(a) cos√x(b) cos|x|(c) cos|x| − 1(d) x2cos√t(e) x2cos√x(f) 2xcos|x|(g) cosx + C(h) cos|x| − 133. ComputeZπ0x sin(x2) dx(a) 0(b) 0.5(c) 2(d) 4(e) π(f)π2(g)π4(h) π −144. Expandx+4(x+1)2by partial fractions.(a)1x+1+x+3(x+1)2(b)2x+1+x+2(x+1)2(c)3x+1+x+1(x+1)2(d)4x+1+x(x+1)2(e)1x+1+3(x+1)2(f)3x+1(g)3(x+1)2(h) impossible since undefined at x = −155. EvaluateZ∞21ln(x)dx(a) Converges, by direct comparison with compari-son function1x2.(b) Diverges, by direct comparison with the compar-ison function1x2.(c) Converges, by direct comparison with the com-parison function ex.(d) Diverges, by direct comparison with the compar-ison function ex.(e) Converges, by direct comparison with the com-parison function1x.(f) Diverges, by direct comparison with the compar-ison function1x.(g) Converges, by direct comparison with the com-parison function e−x.(h) Diverges, by direct comparison with the compar-ison function e−x.66. The region between the curve y = 2√x, 0 ≤ x ≤ 1,and the x-axis is revolved about the x-axis to gener-ate a solid. Compute the volume of this solid.(a) π/2(b) π(c) 2π(d) 5π/2(e) 5π(f) π2(g) π2/2(h) 16π/1577. 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) −1/488. 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 199. 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) none1010. If the Maclaurin series for xex2isPcnxn, find c7.(a) 0(b) 1(c) 1/2(d) 1/6(e) 1/7(f) 1/24(g) 1/7!(h) 1/8!1111. EstimateZ10sin x2dxwith an error less than 0.001.(a) 13/42(b) 1/3(c) 1/42(d) 1/1320(e) 1/7(f) 2/7(g) 1/24(h) 5/241212. Computee−i π/2(a) 0(b) 1(c) −1(d) i(e) −i(f) e(g) −e(h) 1 + i1313. If the Binomial Series for the functionf(x) = (1 + x2)4/3is given byPcnxn, find c4.(a) −1(b) 0(c) 1(d) 1/2(e) 3/4(f) 2/7(g) 4/3(h) 2/91414. Find the area inside one leaf of the four-leaved roser = cos(2θ)(a) π/8(b) π/4(c) π/3(d) π/2(e) π(f) 0.5(g) 1.5(h) 2.515Name: Student ID:15. Find the Taylor Series generated by f(x) = sin x ata = π/2. Show your work. Be sure to include thegeneral term of the series.16Name: Student ID:17Name: Student ID:16. Identify the symmetries of the curver = 1 + cos θand sketch this curve. Show all work.18Name: Student
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