Math 132Midterm Examination 3 – April 4, 20126 multiple choice, 4 long answer. 100 points.General Instructions: Please answer the following, without use of calculators. Youmay refer to a 3x5 card, but no other notes. Part I of the exam is multiple choice,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 geometric series∞Xi=1(−1)i+13iconverges to:(a) 0(b)16(c)14(d)13(e)12(f)23(g)34(h)56(i) 1(j) Does not converge – oscillates.(k) Does not converge – diverges to ∞.2. EvaluateZ8013√xdx.(a) 0(b) 1(c) 2(d) 3(e) 4(f) 5(g) 6(h) 7(i) 8(j) 16(k) ∞3. EvaluateZ∞41x5/2dx.(a) −∞(b) 0(c)120(d)112(e)13(f)12(g)23(h)34(i)54(j) ∞(k) Does not exist/undefined/diverges.4. EvaluateZ2−21x2dx.(a) −2(b) −2 ln 2(c) −1(d) −ln 2(e) 0(f) ln 2(g) 1(h) 2 ln 2(i) 2(j) Does not exist/undefined/diverges.5. EvaluateZπ/20sin 2θ cos θ dθ.(a) 0(b)16(c)14(d)13(e)12(f)23(g)34(h)56(i) 1(j) ∞6. The sequencen4n!converges (as n → ∞) to:(a) −∞(b) −4(c) −2(d) −1(e) 0(f) 1(g) 2(h) 4(i) ∞(j) Does not exist/undefined/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. Exact evaluation of improper integrals and series(a) (6 points) Evaluate∞Xi=12i+ 3i4i.(b) (6 points) For what values of x does∞Xi=0xiconverge? When it converges, whatdoes it converge to?(c) (6 points) EvaluateZ∞0xe−xdx.(d) (6 points) Using partial fractions, evaluate∞Xk=11(k + 1)(k + 2).Name: Id #: Math 1322. Integration techniques(a) (6 points) EvaluateZz + 4z3+ zdz.(b) (6 points) EvaluateZ−9x2− 3x + 6x4− 5x2+ 4dx.(c) (6 points) EvaluateZe3x√1 − e2xdx.(d) (5 points) EvaluateZ101(3 − x2)3/2dx.Name: Id #: Math 1323. Series convergence.Determine whether each of the following series converges or diverges.(a) (6 points)∞Xi=12(i − 1)(i − 2)3(i + 1)(i + 2)(b) (6 points)∞Xi=11i3/2+ sin2i(c) (6 points)∞Xi=12i + ln iName: Id #: Math 1324. Comparison tests for integralsUse a test for convergence for each problem on this page. (Don’t try to find anti-derivatives!)(a) (5 points) Show thatZ∞11x2+√xdx converges.(b) (5 points) Show thatZ101x2+√xdx converges.(c) (1 point) Conclude thatZ∞01x2+√xdx
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