Math 132, Spring 2003Exam 3Name_________________________________________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._____________________________________________________________________________________Part I, Multiple Choice, 5 points/problem: blacken your answers on the answercard.1. A curve is given by 0 , where is some positivecossinœBœ"+ >Cœ #+ >П>Ÿ Ñ +1#constant. The length of the curve is 3. What is the value of ?+A) 1 B) C) D) E) 2"##11F) G) 2 H) I) 3'##$1111#2. A (finite) region in the first quadrant is bounded by the line and the curveCœBCœB B$. What is the volume of the solid obtained by revolving the region around the -axis?A) B) C) D) E) $%( %$( * #% #" )111 11F) G) H) I) J) 11111($$(*#""&&3. A spring is stretched 2 ft beyond its natural length and this requires 6 ft-lbs. of work.How much work would be required to stretch the spring an foot ?additional additionalA) 3 ft-lbs B) 3.5 ft-lbs C) 4 ft-lbs D) 4.5 ft-lbs E) 5 ft-lbsF) 5.5 ft-lbs G) 6 ft-lbs H) 6.5 ft-lbs I) 7 ft-lbs J) 7.5 ft-lbs4. What is the length of the graph of , where ?Cœ B !ŸBŸ#%$$#A) B) C) D) E) "$ "#$( ##"ÈÈ#" # "È$F) G) H) I) 2 J) &*(#94ÈÈ#" # $ "5. A tank 10 ft. tall has the shape of a circular cylinder; its radius is 5 ft. It contains waterto a depth of 2 ft. How much work is done pumping the water up over the top edge of thetank. ÐWater weighs 62.5 lbs/ft ; round your answer to the nearest integer. All answers$are given in ft-lbs).A) 148,742 B) 101,257 C) 98,175 D 95,332 E) 88,357F) 83,753 G) 79,841 H) 77,981 I) 75,369 J) 73,2246. The of a certain solid is pictured. Cross-sections of the solid perpendicular to thebaseC-axis and to the base are quarter-circles. What is the volume of the solid?A) B) C) D) E) 5$11#$#$%&111F) 2 G) H) 4 I) 5 J) 3#""1117. A woman on an island teeming with hungry mosquitos is being bitten at random times.Assume that the random variable “time between bites” has an exponential density\œfunction. If the average time between bites is 1 minute, what is the probability that thetime between bites is two minutes or more? (Round your answer to 4decimal places.)A) 0.0031 B) 0.0132 C) 0.1353 D) 0.1478 E) 0.1798F) 0.2234 G) 0.2437 H) 0.2539 I) 0.2754 J) 0.29718. For the same woman, what is the median time between bites? (All answers are givenin minutes, and rounded to 4 decimal places.)A) 0.0042 B) 0.1397 C) 0.1459 D) 0.3452 E) 0.5512F) 0.6931 G) 0.8734 H) 1.2234 I) 1.3457 J) 1.52439. Suppose all are positive constants. Find (if it exists).+ß ,ß -ß . lim8Ä_+8 , Ð8 Ñ-8 . Ð 8 Ñlnln$#A) B) C) D) E) +, - + +,-. . - -.+.,-F) G) H) I) J) does not exist+. ,-+,+,.--10. For what 's does the series B B B B BÞÞÞœ Ð"Ñ"" " B%"''% %$&( 88œ!_!+1#8 "8+converge?A) 11 B) 11 C) 22 D) 22B ŸBŸ B ŸBŸE) F) G) H) %B% %ŸBŸ% B ŸBŸ"" ""%% %%I) only J) for no values of Bœ! B11. Suppose a random variable has probability density function \for or forfor 0ÐBÑ œ!B!B#B !ŸBŸ"#B "B#ÚÛÜWhat is the mean of ?. \A) 0 B) 0.2 C) 0.4 D) 0.6 E) 0.8F) 1 G) 1.2 H) 1.4 I) 1.6 J) 1.812. Suppose = What is ?!8œ"_8" 8 # $"$Ð"Ñ + œ+++ÞÞÞ Þ +A) B) C) D) E) """"#(&'$F) G) H) I) J) no such exists +""""#$%&13. The sequence defined recursively by for Ö+ ×+œ$+œ( 8"8"8""+œ8has a limit What is ?P !Þ PA) 1 B) 0 C) D) E) ÈÈÈÈ( $( "$(#"F) G) H) I) J) & )' ( &$ & $####(%$ #" # #"ÈÈÈÈÈ14. One of the things you measured in Lab 4 was the change in temperature of atemperature probe that you had . The model for the rate ofpreviously cooled in ice waterchange of temperature of the probe, based on Newton's Law of Cooling, was theXdifferential equation . ( ).X.>œ5ÐXVÑ Notice the “ ” preceding the “ ” ! 55V and were assumed to be constants. In that situation, which of the following statements are true? i) is positiveXV ii) the graph of has the line as a horizontal asymptoteXCœV iii) the graph of is increasingX iv) is negative5A) i), ii) only B) i), iii) only C) i), iv) only D) ii), iii) onlyE) ii), iv) only F) iii), iv) only G) i), ii), iii) only H) i), ii), iv) onlyI) i), iii), iv) only J) all are true Part II, True or False, 1 point each. Blacken your answers on the answer card.15. Suppose is a normal random variable with mean and standard deviation .\ .5Then T Ð\ Ÿ Ñ œ T Ð\ ÑÞ.5 .5 A) True B) False16. Suppose the sequence is increasing and each . Then, for someÖ+ × + Ÿ "(*888œ"_number .Pß + œ Plim8Ä_8 A) True B) False17. The integral test could be used to conclude that the series diverges.!8œ"_8# A) True B) False18. A population is growing according to a logistic model. If the carrying capacity isT5000 individuals and the initial population is 50, then for times near , the population>!grows almost exponentially. A) True B) False19. The equation describes a logistic model (with the equation.T T.> "!!!œ5TÐ& Ñwritten in a slightly nonstandard form). In this model the carrying capacity is 1000. A) True B) FalsePart III These are two “free response” problems (#20, #21) worth a total of 25Àpoints. Write your answers on the test pages. neatly and cross outShow your workirrelevant scratchwork, false starts, etc.Please put your on each of the following pages, since they may be separatednameduring grading. Also, please add your (Discussion Section Letter available on yourexam's front cover sheet) so that we can return papers through discussion sections.Name_______________________________ Discussion Section Letter _________20. a) Does the series converge? Explain why or why not.!8œ"_8"$8 &8(## b) Write down the first 5 terms of a that does converge.geometric series not c) Explain why the Integral Test or apply to the series .does does not!8œ_81e8#Then (using the integral test or some other test) decide whether the series converges
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