Math 132 Spring 2003 Exam 3 Name 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 answer card 1 A curve is given by B cos 0 1 where is some positive C sin constant The length of the curve is 3 What is the value of A 1 F 1 B G 21 C 1 D 1 E 2 H 1 I 1 3 2 A finite region in the first quadrant is bounded by the line C B and the curve C B What is the volume of the solid obtained by revolving the region around the Baxis A 1 B 1 C 1 D F 1 G 1 H 1 I 1 1 E J 1 1 3 A spring is stretched 2 ft beyond its natural length and this requires 6 ft lbs of work How much additional work would be required to stretch the spring an additional foot A 3 ft lbs B 3 5 ft lbs C 4 ft lbs D 4 5 ft lbs E 5 ft lbs F 5 5 ft lbs G 6 ft lbs H 6 5 ft lbs I 7 ft lbs J 7 5 ft lbs 4 What is the length of the graph of C B where B A B C F G 9 4 H D I 2 E J 5 A tank 10 ft tall has the shape of a circular cylinder its radius is 5 ft It contains water to a depth of 2 ft How much work is done pumping the water up over the top edge of the tank Water 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 88 357 F 83 753 G 79 841 H 77 981 I 75 369 J 73 224 6 The base of a certain solid is pictured Cross sections of the solid perpendicular to the C axis and to the base are quarter circles What is the volume of the solid 1 A 1 B F 2 G 1 C 1 H 41 D 1 I 51 E 51 J 3 7 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 function If the average time between bites is 1 minute what is the probability that the time between bites is two minutes or more Round your answer to 4 decimal places A 0 0031 B 0 0132 C 0 1353 D 0 1478 E 0 1798 F 0 2234 G 0 2437 H 0 2539 I 0 2754 J 0 2971 8 For the same woman what is the median time between bites All answers are given in minutes and rounded to 4 decimal places A 0 0042 B 0 1397 C 0 1459 D 0 3452 E 0 5512 F 0 6931 G 0 8734 H 1 2234 I 1 3457 J 1 5243 8 ln 8 8 8 ln 8 9 Suppose all are positive constants Find lim B C D F G H I A 10 For what B s does the series B B if it exists E J does not exist B B 8 8 1 B 8 8 converge A 1 B 1 B 1 B 1 C 2 B 2 E B F B G I B only J for no values of B B 11 Suppose a random variable has probability density function D 2 B 2 H B for B or B B for B 0 B B for B What is the mean of A 0 B 0 2 C 0 4 D 0 6 E 0 8 F 1 G 1 2 H 1 4 I 1 6 J 1 8 12 Suppose 8 8 What is 8 A F B G C H D I E J no such exists 13 The sequence 8 defined recursively by has a limit P What is P A 1 F B 0 G 8 8 for 8 C D E H I J 14 One of the things you measured in Lab 4 was the change in temperature of a temperature probe that you had previously cooled in ice water The model for the rate of change of temperature X of the probe based on Newton s Law of Cooling was the differential equation X 5 X V Notice the preceding the 5 5 and V were assumed to be constants In that situation which of the following statements are true i X V is positive ii the graph of X has the line C V as a horizontal asymptote iii the graph of X is increasing iv 5 is negative A i ii only B i iii only C i iv only D ii iii only E ii iv only F iii iv only G i ii iii only H i ii iv only I 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 5 Then T 5 T 5 A True B False 16 Suppose the sequence 8 8 is increasing and each 8 Then for some number P lim 8 P 8 A True B False 17 The integral test could be used to conclude that the series 8 diverges 8 A True B False 18 A population T is growing according to a logistic model If the carrying capacity is 5000 individuals and the initial population is 50 then for times near the population grows almost exponentially A True B False T 19 The equation T 5T describes a logistic model with the equation written in a slightly nonstandard form In this model the carrying capacity is 1000 A True B False Part III These are two free response problems 20 21 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 s front cover sheet so that we can return papers through discussion sections Name Discussion Section Letter 20 a Does the series 88 8 converge Explain why or why not 8 b Write down the first 5 terms of a geometric series that does not converge c Explain why the Integral Test does or does not apply to the series 8 1 8 e8 Then using the integral test or some other test decide whether the series converges or diverges Name Discussion Section Letter Also please add your Discussion Section Letter available on your exam s front cover sheet above so that we can return papers through discussion sections …
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