NAME:1 /15 2 /15 3 /10 4 /10 5 /106 /15 7 /5 8 /20 9 /10 10 /10 T /120MATH 152 H/M (Spring 2005) Final Exam, May 20thNo 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−1x. Find g0(π/4).(b) Find sinh−1(3).(2) [15 pts] Calculate the following integrals(a)Rsin3θ cos2θ dθ2(b)Rdx√2x−x2(c)R7x2−6x+4x3−2x2+xdx3(3) [10 pts] Determine whether the improper integralR∞1x√x6+1is convergent or divergent.(4) [10 pts] The speedometer reading (v) on a car was observed at 1-minute intervals and recorded in thechart below. Use Simpson’s Rule to estimate the distance travelled by the car. [You should leave youanswer as a sum of numbers.]t (min) v (miles/hr)0 401 422 453 494 524(5) [10 pts] (a) Explain why the functionf(x) =(12sin x if 0 ≤ x ≤ π,0 if x < 0 or x > πis a probability density function.(b) Calculate the mean of this probability density function.5(6) [15 pts] Determine whether the following series are absolutely convergent, conditionally convergent, ordivergent.(a)P∞n=1(−1)n−1√nn+1(b)P∞n=11·3·5···(2n−1)5nn!6(c)P∞n=1tan(1√n)(7) [5 pts] Prove that if the series∞Pn=1anis convergent then limn→∞an= 0.7(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).8(c) How good an approximation is T2(14) to ln12?(9) [10 pts] One version of Zeno’s paradox concerns a person who is walking in a straight line from pointA to point B. If you are currently at A and wish to move to B, you must first travel half the distanceto 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 remainingdistance 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 BZeno’s way is indeed 100 seconds.9(10) [10 pts] This question asks you to prove a simplified version of the Ratio Test. Specifically, supposethat anis a sequence with an> 0 for all n and thatan+1an<12for all n.(a) Show that an< a0(12)n(b) Using (a) show that∞P1anconverges.Pledge: I have neither given nor received aid on this