NAME 1 15 2 15 3 10 4 10 5 10 6 15 7 5 20 9 10 10 10 T 8 MATH 152 H M Spring 2005 Final Exam May 20th No calculators books or notes Show all work and give complete explanations for all your answers This is a 2 hour exam It is worth a total of 120 points 1 15 pts a Let g be the inverse of the function f x ln x tan 1 x Find g 0 4 120 b Find sinh 1 3 2 15 pts Calculate the following integrals a R sin3 cos2 d b R dx 2x x2 c R 7x2 6x 4 x3 2x2 x dx 3 10 pts Determine whether the improper integral R 1 x x6 1 is convergent or divergent 4 10 pts The speedometer reading v on a car was observed at 1 minute intervals and recorded in the chart below Use Simpson s Rule to estimate the distance travelled by the car You should leave you answer as a sum of numbers t min 0 1 2 3 4 v miles hr 40 42 45 49 52 5 10 pts a Explain why the function f x 1 2 0 sin x if 0 x if x 0 or x is a probability density function b Calculate the mean of this probability density function 6 15 pts Determine whether the following series are absolutely convergent conditionally convergent or divergent n 1 n n 1 1 n 1 a P b P n 1 1 3 5 2n 1 5n n c P n 1 tan 1n 7 5 pts Prove that if the series P n 1 an is convergent then lim an 0 n 8 20 pts Let f x ln 1 2x a Calculate the Taylor s series for f about x 0 b Find the interval of convergence for the Taylor s series in a c How good an approximation is T2 14 to ln 12 9 10 pts One version of Zeno s paradox concerns a person who is walking in a straight line from point A to point B If you are currently at A and wish to move to B you must first travel half the distance to B and then half the remaining distance and then half the remaining distance and so on ad infinitum It seems like you ll never reach B since no matter where you are you always have half the remaining distance to travel Suppose the distance from A to B is 100 meters and that you travel at a speed of 1 meter per second Use an infinite series to resolve Zeno s paradox i e show that the total time taken to travel from A to B Zeno s way is indeed 100 seconds 10 10 pts This question asks you to prove a simplified version of the Ratio Test Specifically suppose that an is a sequence with an 0 for all n and that an 1 12 for all n an a Show that an a0 21 n b Using a show that P an converges 1 Pledge I have neither given nor received aid on this exam Signature