Mathematics 3 Final Exam 0 May 03 Directions: This exam should consist of twelve multiple choice questions and four handgraded questions. Multiple choice questions are worth five points apiece. The first two handgraded questions are worth ten points each. The last two hand-graded questions are worth fifteen points each. The maximum possible score is 0 points. Write your student ID number carefully at the top of your answer card, using one line per digit. Print your name on the top of the card in the space provided. Mark your answer card with a PENCIL by filling in the appropriate box. On the hand-graded sheet, write your name and student ID number in the spaces provided. You may not use a calculator. You may not use any written aids.
Problem : Let f(x) = sin x. What is f (0)? A. B. ln C. D. ln E. 0 F. ln G. H. ln I. J. This function is not differentiable at x = 0. Problem : 00 Consider the following quantity: [ ( + ) 4 ( + + ) 4 ( + + + 99 ) 4 ( + + 00 ) ] 4 00 00 00 00 This quantity is approximately equal to which of the following? A. B. C. D. E. F. G. H. 0 0 0 0 x 4 dx x 4 dx 00 x4 dx 00 x4 dx + x 4 dx + x 4 dx 00 ( + x4 )dx 00 ( + x4 )dx
Problem 3: Consider the function f(x) = 3( x) 3 on the interval [0, ]. What is the maximum value attained by f on this interval? A. 0 B. C. D. 3 E. 4 F. 5 G. 6 H. 7 I. 8 J. 9 Problem 4: Consider the curve x +x y +4y 5 +y =. What is the slope of the tangent line to this curve at the point (, 0)? A. - B. C. - D. E. -3 F. 3 G. -4 H. 4 I. -5 J. 5 3
Problem 5: Suppose that the acceleration of a particle at time t is given by 6t + and at time t = 0 the particle has velocity and position 0. Which of the following gives the position of the particle for any time t? A. 4t 3 6t B. 4t 3 6t C. 4 D. 4t 3 + t + E. 3t 4 t F. t 3 t t G. 3t 3 + t + H. t 3 + t + t I. t 3 t + J. t 3 Problem 6: Suppose that f is a continuous function whose domain is (, ). Beyond this, you have only the following data: x f(x) 3-4 5-6 - 7-8 9 - What is the smallest number of zeros f could have? A. 9 B. 8 C. 7 D. 6 E. 5 F. 4 G. 3 H. I. J. 0 4
Problem 7: Compute the following limit: A. -4 B. -3 C. - D. - E. 0 F. G. H. 3 I. 4 J. The limit does not exist. lim x 0 + x 3 + x cos x + sin x. Problem 8: Which of the following is a horizontal asymptote of + 4 x x? A. y = 4 B. y = 3 C. y = D. y = E. y = 0 F. y = G. y = H. y = 3 I. y = 4 J. This function has no horizontal asymptotes. 5
Problem 9: Compute the following: 0 x 3 x + dx A. 0 B. C. D. 3 E. 4 F. 5 G. 6 H. 7 I. 8 J. 9 Problem 0: Compute the following: 0 9t (t 3 + ) / dt. Note: 3/ =, 4 3/ = 8, 9 3/ = 7, 6 3/ = 64, and so on. A. 45 B. 46 C. 47 D. 48 E. 49 F. 50 G. 5 H. 5 I. 53 J. 54 6
Problem : π/ ( + e sin x ) cos xdx A. 0 B. C. e π D. e E. e F. πe G. e + π H. + e I. e π/ J. e + π 0 Problem : What is F (0)? A. 0 B. C. D. 3 E. 4 F. 5 G. 6 H. 7 I. 8 J. 9 Suppose that F (x) = x 4 cos t dt. 7
Math 3 Final Exam: Hand-Graded Problems Student Name: Student ID #: Problem 3: Suppose that f is a continuous function and let F (x) = x 0 f(t)dt. Suppose that the graph of F (x) on the interval [ 5, 7] is given below. How many zeros does f have in the interval [ 5, 7]? Explain your answer. Be sure to refer to any theorems you might be using. Remember, the graph above is of F (x). 8
Math 3 Final Exam: Hand-Graded Problems Student Name: Student ID #: Problem 4: Suppose you are driving along Interstate 55 from St. Louis to Chicago. The distance you travel along this highway is 300 miles. Immediately upon your arrival in Chicago, you are pulled over by the police at precisely 9:30PM because one of your lights is broken. During your conversation, you tell the officer you didn t have time to fix your light because you didn t get out of your last class at Washington University in St. Louis until a little after 5:30PM. The officer says, Well, I won t give you a ticket for your broken light, but I m going to have to give you a ticket for speeding. You can blame my calculus professor at WashU for teaching me the mean value theorem. Assuming the speed limit is 65 miles per hour, is the officer mathematically justified for writing this ticket? Explain your answer. Clearly, you are justified in blaming a WashU calculus professor either way. 9
Math 3 Final Exam: Hand-Graded Problems Student Name: Student ID #: Problem 5: Let f(x) = x4 4 x3 9x. Note that f(x) = 0 if and only if x = 0, ± 0 (this is approximately ± 6.35). (a) (4 points) Compute f (x) and f (x). (b) (4 points) Identify the local extrema of f, if there are any. (c) (4 points) Identify the inflection points of f, if there are any. (d) (3 points) Sketch the graph of f, labeling your axes appropriately. 0
Math 3 Final Exam: Hand-Graded Problems Student Name: Student ID #: Problem 6: Let f(x) = x x. Note that f(x) = 0 if and only if x = ± 5 (this is approximately 0.5 ±.8). (a) (3 points) Compute f (x) and f (x). (b) (3 points) Identify the local extrema of f, if there are any. (c) (3 points) Identify the inflection points of f, if there are any. (d) (3 points) Determine any vertical, horizontal, or oblique asymptotes which f might have. (e) (3 points) Sketch the graph of f, labeling your axes appropriately.