Math 132Final Examination – May 4, 20126 multiple choice, 4 long answer. 100 points.General Instructions: Please answer the following, without use of calculators. Youmay refer to up to four 3x5 cards, but no other notes. Part I of the exam is multiplechoice, while Part II is long answer.Part I Instructions: If you do not have a pencil to fill out your answer card, please askto borrow one from your proctor. Write your Student ID number on the six blank lineson the top of your answer card, and shade in the corresponding bubbles to the rightof each digit.Fill in the bubble corresponding to each of the following 6 questions. Each is worth 4points. On Part I, no partial credit will be given.1. The Taylor series for f(x) =11 − x3(around 0) is(a)∞Xk=0xkk!(b)∞Xk=0x3kk!(c)∞Xk=0xk(3k)!(d)∞Xk=0x3k(e)∞Xk=3xk(f)∞Xk=3xkk!(g)∞Xk=3x3k(h) None of the above.2. Evaluate∞Xi=216 ·−34i.(a) 0(b)47(c) 1(d) 2(e) 3(f) 4(g) 5(h)367(i) Does not converge – oscillates.(j) Does not converge – diverges to ∞.3. Evaluate limn→∞nXi=18i3n4by recognizing it as a Riemann sum.(a) 0(b)12(c) 1(d)32(e) 2(f)52(g) 3(h)72(i) 4(j)92(k) Diverges to ∞4. Which of the following series can the Alternating Series test be used on?I.∞Xi=0(−1)ii2+ 1II.∞Xi=0(−1)ii cos2iIII.∞Xi=1(−1)iiei(a) None of them.(b) I only.(c) II only.(d) III only.(e) I and II only.(f) I and III only.(g) II and III only.(h) All of I, II, and III.5. Evaluate limi→∞2i+1+ i22i+ 3i3.(a) 0(b)13(c)25(d)12(e)23(f) 1(g)43(h)32(i) 2(j) 3(k) Diverges to ∞.6. EvaluateZπ0x sin(x2) dx.(a) −2π(b) −4(c) −π(d) −2(e) 0(f) 1(g) 2(h) π(i) 4(j) 2π(k) Diverges.Name: Id #: Math 132Part II Instructions: Answer the following on the exam sheet, showing all your work.Correct answers without correct supporting work may not receive full credit. You mayuse the back of each page for additional answer space (please clearly indicate if youhave done so), or scratch work.Please put your name and student id number on each page of Part II now.1. Calculations(a) (7 points) Find all solutions to the differential equation y0=y1 − x(assume x < 1).(b) (6 points) Find an upper bound for |2e−x+ 3ex+ 4 sin x + 5 cos x| on the interval(−1, 3).(c) (6 points) Using a power series, find f(100)(0) and f(101)(0) for f(x) = ex2.Name: Id #: Math 1322. Integrals(a) (5 points) Set up an integral for the area of the surface obtained by rotating thecurve y = e−2xaround the x-axis, for x between 0 and ∞.(You need not evaluate the integral in question.)(b) (6 points) Show that the improper integralZ∞011 + x4dx converges.(c) (8 points) Find the volume of the solid obtained by rotating the region below thecurve y =13x + 1about the x-axis for 0 ≤ x < ∞ .Name: Id #: Math 1323. Series and power series. Use the back if you need additional space.(a) (7 points) Using partial fractions, find a power series representation for4(x − 1)(x − 3).(b) (8 points) Find the radius and interval of convergence for the power series∞Xi=02ii2+ 2xi.(c) (6 points) Does the series∞Xi=0(−1)ii3/2− 1converge absolutely, converge conditionally,or diverge?Name: Id #: Math 1324. Integrating cos x2.(a) (1 point) We have discussed that cos x2has no closed form antiderivative. In 1-2sentences, explain what this means.(b) (6 points) Using the Fundamental Theorem of Calculus and an appropriate defi-nite integral, give an antiderivative of cos x2.(c) (6 points) Using a Taylor series expansion, give a power series representation ofan antiderivative of cos x2.(d) (2 points) Using part (c), find a series representingZ10cos
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