Math 132, Fall 2003Exam 3Name: ID#No calculators with a CAS are allowed. Be sure your calculator is set for ”radians”,not ”degrees”, if you do any calculus computations with trig functions. For Parts Iand II, please mark your answer on the answer card. For Part III, please solve theproblems in the space provided.Part I, Multiple Choice, 5 points/problem:1. Suppose you recently bought a new television. The manufacturer claims that on average, thismodel lasts 5 years. Assuming an exponential probability density function, what is the probabilitythat your television will last at least 5 years?A) .001 B) .0067 C) 0.2000 D) .2874 E) .3216F) .3679 G) .4138 H) .5000 I) .7358 J) 1.839412. Suppose 3 ft-lb of work are required to stretch a spring from 2 ft to 4 ft beyond its naturallength. How far beyond the spring’s natural length will a 5 lb weight stretch the spring? (Allanswers are given in units of feet.)A) 20 B) 18.75 C) 15 D) 10 E) 8F) 6.25 G) 5 H) 3.33 I) 2 J) 1.53. Which of the following infinite series converge?(I)∞Xn=21n2− 1(II)∞Xn=21n2(III)∞Xn=21n2+ 1A) none B) I only C) II only D) III onlyE) I and II F) I and III G) II and III H) all24. The first term of a geometric series is 3, and the fourth term is8164. Find the sum of the series.A)34B) −11181C)2764D) −32E)19237F)12G) 2 H) 4 I) 6 J) 1235. Let S =∞Xn=11n3= 1 +123+133+143+ ···. (We know this series converges by the Integral Test.)Suppose we approximate S using the partial sum S5. Then S −S5= R5is the error. Use the errorestimates associated to the Integral Test to determine what is the best we can say about the error.A)1106≤ R5≤1100B)1104≤ R5≤1100C)192≤ R5≤186D)172≤ R5≤150E)168≤ R5≤164F)160≤ R5≤152G)146≤ R5≤142H)142≤ R5≤138I)134≤ R5≤130J)124≤ R5≤12046. A uniform chain 10 ft. long and weighing 32 lb is lying in a coil at ground level. Find the workrequired to raise the chain so it hangs vertically with the bottom of the chain 2 ft above the groundlevel. Answers are given in units of ft-lb. [Hint: You might want to first consider the case whenthe chain is raised so the bottom of the chain is at ground level.]A) 100 B) 150 C) 314 D) 512 E) 1664F) 1600 G) 384 H) 320 I) 224 J) 1607. A certain series∞Xn=1anhas the following partial sums:S1= 3S2= 1S3= 4S4= 1S5= 5Find a4.A) −3 B) −2 C) 1 D) 2 E) 3F) 4 G) 5 H) 8 I) 9 J) 1458. Find the sum of the series∞Xn=1nn + 1−n + 1n + 2.A) 0 B) −12C)32D) 1 E)76F) −32G) −16H)56I)12J) diverges69. Let S =∞Xn=1(−1)n+1(2n)!. The partial sumSn=12!−14!+16!−18!+ ··· +(−1)n+1(2n)!.What is the smallest n we can find to guarantee |S − Sn| < .0001?A) n = 10 B) n = 9 C) n = 8 D) n = 7 E) n = 6F) n = 1 G) n = 2 H) n = 3 I) n = 4 J) n = 510. Determine a so that the functionf(t) =(a2− t2if 0 ≤ t ≤ a0 otherwiseis a probability density function.A) a =3r34B) a =r13C) a =23D) a =32E) a =3r32F) a = 0 G) a =3223H) a =r116I) a =3r43J) a = 1711. Let b and c be positive constants. Determine whether the sequencean=22n+14n−1+bn2+ ln2n2cn2+√nconverges or diverges. If it converges, then please find its limit.A) 0 B)bc+12C) 8 +b + ln 2cD) 8 +bcE) 8F)bcG) 1 H) 8 + ln 2 I)12J) diverges812. For what values of a > 0 will the series∞Xn=1nln aconverge?A) converges for all a > 0 B) a > e C) a > 2 D) a > 1 E) a = 1F) 0 < a < 1 G) 0 < a <1eH) 0 < a <12I) 0 < a < e J) diverges for all a13. Find the mean of the probability density functionf(t) =(t16if 2 ≤ t ≤ 60 otherwise.A)2083B) 1 C)92D)98E) 0F)√20 G)133H) 2√5 I)r138J)1169Part II: True/False (2 points each)14. If∞Xn=1anconverges (an6= 0), then∞Xn=11anconverges.A) True B) False15. If {an} and {bn} are divergent sequences, then the sequence {anbn} is divergent.A) True B) False16. The Integral Tests shows that∞Xn=11n2= 1.A) True B) False17. Suppose f (t) is a probability density function and F (x) its cumulative distribution function.Then limt→∞f(t) = 0 and limx→∞F (x) = 1.A) True B) False18. We can use the Alternating Series Test to conclude that∞Xn=1cos (nπ)n2converges.A) True B) False10Part III: These are the ”free response” problems worth a total of 25 points. Write youranswers on the test pages. Show your work neatly and cross out irrelevant scratch-work, false starts, etc.Please put your NAME on each of the following pages, since they may be separatedduring grading. Also, please add your Discussion Section Letter (available on yourexam front cover) on each page so that we can return these pages in your discussionsection.Name: Discussion Section:19. Determine whether each of the following series converges or diverges. Be sure to state whichtest you are using and to verify the hypotheses of each test.a)∞Xn=114n− 2nb)∞Xn=11 − 2n210n2+ 111c)∞Xn=12 + cos nn4+ n12Name: Discussion Section:20. For each of the following, please write an integral that gives the solution for each problem.Please do not evaluate the integrals.a) What is the arc length of the curve given by x = 2et, y = 2e−t, 0 ≤ t ≤ 2.b) How much work does it take to pump half of the water from a full right circular cylindrical tankof radius 5 meters and height 10 meters to a level 4 meters above the top of the tank. [Recall thatthe density of water is 1000 kg/m3.]13c) The volume of a solid that has a base that is a circle of radius R in the x − y plane. Crosssections of the solid, perpendicular to the x-axis, are
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