Name Math 1b Final Exam Tuesday May 23 2006 Please circle your section Rina Anno 10 11 MWF Thomas Judson 11 12 MWF Li Sheng Tseng 10 11 MWF Robin Gottlieb 10 11 30 TTh Problem Number 1 2 3 4 5 6 7 8 9 10 11 12 13 Total Robert Strain 11 12 MWF Robin Gottlieb 11 30 1 TTh Possible Score Points 9 4 7 8 8 15 8 4 9 8 6 6 8 100 Directions Please Read Carefully You have three hours to take this exam Make sure to use correct mathematical notation To receive full credit on a problem you will need to justify your answers carefully unsubstantiated answers will receive little or no credit Please be sure to write neatly illegible answers will receive little or no credit If more space is needed use the back of the previous page to continue your work Be sure to make a note of this on the problem page so that the grader knows where to find your answers Calculators are not allowed Good Luck 1 1 9 points Evaluate the following integrals Z 5x 7 dx a x 1 x 2 Z b 1 x arctan x dx 0 Z c 2 1 dx x ln x 2 2 2 4 points Suppose that you wish to model a population with a differential equation of the form dP dt f P where P t is the population at time t Experiments have been performed on the population that give the following information The population at P 0 remains constant A population of 0 P 20 will decrease A population of P 20 does not change A population of 20 P 100 increases A population of P 100 will decrease Which of the following differential equations best models this population Circle the correct answer a dP P 20 P 100 dt b dP P 20 P P 100 dt c dP P P 20 100 P dt d dP P 20 P 100 P dt e dP 20 P P 100 dt 3 3 7 points A bag of sand originally weighing 144 lb is lifted at a constant rate As it rises sand leaks out at a constant rate The sand is half gone by the time the bag has been lifted 18 ft a How many pounds of sand leak out of the bag per foot as the bag is lifted b How much work was done in lifting the bag 18 ft To receive full credit for your work indicate clearly what your variable is and what you are considering to be zero 4 graphs final 1b nb 4 8 points Let R be the region bounded by the curves y f x and y g x shown in the graph below y 4 H 3 2L f HxL 2 4 2 gHxL 2 4 x 2 H2 3L 4 a Write a definite integral that will give the area of the region R b Write a definite integral that will give the volume of the solid generated when the region R is revolved about the horizontal line y 4 c If the base of a solid V is the region R and the cross sections of the solid perpendicular to the x axis are squares write a definite integral that will give the the volume of V 5 5 8 points A resort town is laid out along the seashore in the shape of a semicircle of radius 3 miles with the diameter of the semicircle bordering the ocean People want to live close graphs final 1b nb to the center of the town indicated by the solid black disk The density of the population individuals per square mile at a distance of x miles from the center of town is given by x 600 200x Resort Ocean a Write a Riemann sum that approximates the total population of the resort b Use your answer from part a to write a definite integral that represents the total population of the town and evaluate the integral 6 6 15 points In an isolated region of the Canadian Northwest Territories a population of arctic wolves and a population of silver foxes compete for resources The two species have a common limited food supply which consists mainly of mice If x t the population of arctic wolves in thousands y t the population of silver foxes in thousands the interaction of the two species can be modeled by the following system of differential equations dx x x2 xy dt dy 3 1 y y 2 xy dt 4 2 a Find the nullclines of the system for x 0 and y 0 Axes are provided on the next page b Find all of the equilibrium points for x 0 and y 0 7 dx x x2 xy dt dy 3 1 y y 2 xy dt 4 2 c What happens to the arctic wolf population in the absence of silver foxes What happens to the silver fox population in the absence of arctic wolves d Sketch and label the nullclines for x 0 and y 0 Be sure to indicate the graphs final 1b nb direction of the solution on the nullclines and in the regions bounded by the nullclines y 2 1 5 1 0 5 0 5 1 1 5 2 x e Sketch the solution trajectory with the initial conditions x 0 0 5 and y 0 1 5 indicating the direction of your solution curve 8 7 8 points A dosage d of a drug is given daily at t 0 1 2 3 days The drug decays exponentially at a rate r in the blood stream Thus the amount in the bloodstream after n 1 doses is d de r de 2r de nr a Find the level of the drug after an infinite number of doses That is find d de r de 2r de nr b If r 0 1 what dosage is needed to maintain a drug level of 2 9 8 4 points The following polynomials are second degree Taylor polynomials for functions whose graphs are given below Match each Taylor polynomial with the appropriate graph a T2 x 2 x 1 2 x 1 2 5 b T2 x 2 2 x 1 x 1 2 2 c T2 x 4 x 1 9 x 1 2 8 13 d T2 x 3 x 1 x 1 2 3 6 Untitled 1 Untitled 1 i ii 1 iii y y 4 3 2 4 4 3 3 2 2 1 1 1 Untitled 1 1 2 3 4 x 4 3 2 1 1 1 1 2 2 3 3 graphs final 1b nb 4 i iii 2 4 x 1 4 4 3 3 2 2 1 1 1 3 y y 3 2 4 ii 4 iv 1 2 3 4 x 4 3 2 1 1 1 1 2 2 3 3 4 iv 10 4 2 3 4 x 9 9 points functions x a 1 x Find a power series representation at x 0 for each of the following …