Math 132 Fall 2003 Exam 3 Name 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 I and II please mark your answer on the answer card For Part III please solve the problems 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 this model lasts 5 years Assuming an exponential probability density function what is the probability that your television will last at least 5 years A 001 F 3679 B 0067 G 4138 C 0 2000 H 5000 D 2874 I 7358 1 E 3216 J 1 8394 2 Suppose 3 ft lb of work are required to stretch a spring from 2 ft to 4 ft beyond its natural length How far beyond the spring s natural length will a 5 lb weight stretch the spring All answers are given in units of feet A 20 F 6 25 B 18 75 G 5 C 15 H 3 33 D 10 I 2 E 8 J 1 5 3 Which of the following infinite series converge I X n 2 A none E I and II 1 2 n 1 B I only F I and III X 1 II n2 III n 2 C II only G II and III 2 X n 2 D III only H all n2 1 1 4 The first term of a geometric series is 3 and the fourth term is A 3 4 B F 1 2 G 2 111 81 C 27 64 H 4 D I 6 3 3 2 E 81 64 Find the sum of the series 192 37 J 12 X 1 1 1 1 1 3 3 3 We know this series converges by the Integral Test 3 n 2 3 4 n 1 Suppose we approximate S using the partial sum S5 Then S S5 R5 is the error Use the error estimates associated to the Integral Test to determine what is the best we can say about the error 5 Let S A 1 1 R5 106 100 B D 1 1 R5 72 50 E G 1 1 R5 46 42 J 1 1 R5 24 20 H 1 1 R5 104 100 1 1 R5 68 64 1 1 R5 42 38 4 C 1 1 R5 92 86 F 1 1 R5 60 52 I 1 1 R5 34 30 6 A uniform chain 10 ft long and weighing 32 lb is lying in a coil at ground level Find the work required to raise the chain so it hangs vertically with the bottom of the chain 2 ft above the ground level Answers are given in units of ft lb Hint You might want to first consider the case when the chain is raised so the bottom of the chain is at ground level A 100 B 150 C 314 D 512 E 1664 F 1600 G 384 H 320 I 224 J 160 7 A certain series X an has the following partial sums n 1 S1 3 S2 1 S3 4 S4 1 S5 5 Find a4 A 3 F 4 B 2 G 5 C 1 H 8 D 2 I 9 5 E 3 J 14 X n n 1 8 Find the sum of the series n 1 n 2 n 1 A 0 F 3 2 B 1 2 C 3 2 D 1 G 1 6 H 5 6 I 1 2 6 E 7 6 J diverges 9 Let S X 1 n 1 n 1 2n The partial sum Sn 1 1 1 1 1 n 1 2 4 6 8 2n What is the smallest n we can find to guarantee S Sn 0001 A n 10 F n 1 B n 9 G n 2 C n 8 H n 3 D n 7 I n 4 E n 6 J n 5 10 Determine a so that the function a2 t2 f t 0 if 0 t a otherwise is a probability density function A a r 3 F a 0 3 4 B a r 1 3 2 3 3 G a 2 2 C a 3 H a 7 r 3 D a 2 11 6 I a r 3 E a 4 3 r 3 J a 1 3 2 11 Let b and c be positive constants Determine whether the sequence 2 ln 2n2 bn 2n 1 2 an n 1 4 cn2 n converges or diverges If it converges then please find its limit A 0 F b c B b 1 c 2 G 1 C 8 b ln 2 c D 8 H 8 ln 2 I 8 1 2 b c E 8 J diverges 12 For what values of a 0 will the series X nln a converge n 1 A converges for all a 0 B a e C a 2 D a 1 E a 1 F 0 a 1 1 G 0 a e 1 H 0 a 2 I 0 a e J diverges for all a 13 Find the mean of the probability density function t if 2 t 6 f t 16 0 otherwise A F 208 3 20 B 1 G 13 3 C 9 2 H 2 5 D I 9 8 r 9 E 0 13 8 J 1 16 Part II True False 2 points each 14 If X an converges an 6 0 then n 1 A True X 1 converges an n 1 B False 15 If an and bn are divergent sequences then the sequence an bn is divergent A True B False X 1 1 16 The Integral Tests shows that n2 n 1 A True B False 17 Suppose f t is a probability density function and F x its cumulative distribution function Then lim f t 0 and lim F x 1 t A True x B False 18 We can use the Alternating Series Test to conclude that X cos n n 1 A True B False 10 n2 converges Part III These are the free response problems worth a total of 25 points Write your answers on the test pages Show your work neatly and cross out irrelevant scratchwork false starts etc Please put your NAME on each of the following pages since they may be separated during grading Also please add your Discussion Section Letter available on your exam front cover on each page so that we can return these pages in your discussion section Name Discussion Section 19 Determine whether each of the following series converges or diverges Be sure to state which test you are using and to verify the hypotheses of each test a X n 1 4n 1 2n X 1 2n2 b 10n2 1 n 1 11 c X 2 cos n n 1 n4 n 12 Name Discussion Section 20 For each of the following please write an integral that gives the solution for each …
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