Download Mathematics 106 Final Exam: Integrals, Series, and Differential Equations and more Exams Calculus in PDF only on Docsity! Name: ___________________________ Section (circle one):MWRF 11:00-11:55 MWRF 12:05-1:00 Mathematics 106 Final Exam December 11, 2001 Problem Possible Actual 1 12 2 12 3 10 4 12 5 10 6 10 7 12 8 10 9 12 Total 100 You must show all work to receive credit. Calculators are permitted. Give exact answers (ln 5, e2) unless requested otherwise. 1. Evaluate each of the following. a) โซ โ โ 0 5 dxxe x b) โซ โ e xx dx ln c) โซ dxe x 2. A cylindrical tank of length 12 feet and radius 4 feet is buried on its side 5 feet below ground. It is filled to a height of 3 feet with gasoline, which weighs 42 pounds/cubic foot. Write (but do NOT evaluate) an integral equal to a) the volume of the gasoline in the tank b) the work done in pumping all the gasoline to ground level c) the fluid force exerted by the gasoline on one of the circular ends of the tank 7. )(tx and )(ty give the populations (in 1000s) of two interacting animal populations as functions of time. They interact according to the system of differential equations below. xyx dt dx 2 12 โ= xyyy dt dy 6 1)3( +โ= a) In the absence of population y, how does population x behave? What happens to x as โโt ? b) In the absence of population x, how does population y behave? What happens to y as โโt ? c) Characterize the interaction between the two populations. d) Find the null-clines of this system. e) Sketch the null-clines and indicate with an arrow the direction of the trajectories in each sector. f) Sketch on these same axes the trajectory that starts at the point (10,2). 8. You deposit money at a steady rate of $4000 per year into an account that earns 8% continuously compounded interest. Let )(tB be the balance in the account t years after you begin deposits. a) Write a differential equation whose solution is B(t). b) Solve this differential equation. c) This problem can also be solved using an integral. Write (but do NOT evaluate) an integral giving the value of this stream of money t years after you begin deposits. (The value of this integral should turn out to equal the B(t) you found in part (b). ) 9. a) A lake initially has a volume of 1010 liters of pure water. Acid rain containing .03 mg of pollutants per liter falls into the lake at a rate of 610 liters per year and is well-mixed into the lake. A stream draws away 610 liters per year of the mixture of water and pollutants in the lake. Write (but do NOT solve) a differential equation whose solution is P(t), the amount (in mg) of pollutant in the lake t years later. b) Find the exact value of i) 10 + 5 + 2.5 + 1.25 + โฆ ii) 1 โ 1 + โ+โ+โ 720 1 120 1 24 1 6 1 2 1 โฆ iii) )0()13(f if )2cos()( 45 xxxf = c) What is the maximum possible error if you approximate the value of e using the 1000th degree Taylor polynomial for xexf =)( about 0=x ? You may use the fact that e < 3.
