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1 MTH 33 Solutions to Final Exam May, 8 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response questions. Write your answers clearly! Include enough steps for the grader to be able to follow your work. Don t skip limits or equal signs, etc. Include words to clarify your reasoning. Do first all of the problems you know how to do immediately. Do not spend too much time on any particular problem. Return to difficult problems later. If you have any questions please raise your hand and a proctor will come to you. You will be given exactly minutes for this exam. Remove and utilize the formula sheet provided to you at the end of this exam. ACADEMIC HONESTY Do not open the exam booklet until you are instructed to do so. Do not seek or obtain any kind of help from anyone to answer questions on this exam. If you have questions, consult only the proctor(s). Books, notes, calculators, phones, or any other electronic devices are not allowed on the exam. Students should store them in their backpacks. No scratch paper is permitted. If you need more room use the back of a page. Anyone who violates these instructions will have committed an act of academic dishonesty. Penalties for academic dishonesty can be very severe. All cases of academic dishonesty will be reported immediately to the Dean of Undergraduate Studies and added to the student s academic record. I have read and understand the above instructions and statements regarding academic honesty:. SIGNATURE Score: /8 α Page of

2 MTH 33 Solutions to Final Exam May, 8 Standard Response Questions. Show all work to receive credit. Please BOX your final answer.. (6 points) How much work does it take to pump the water from a full upright circular cylindrical tank of radius 5 ft and height 3 ft to a level of ft above the top of the tank? (Water weighs 63 lbs/ft 3.) Solution: W = 3 63 (4 y)(π5 ) dy ] 3 = 63 5π [4y y [ = 63 5π 4(3) 9 ] = 63 5π 75 ft lbs. (6 points) Solve the initial value problem dy dx = y + x with y() =. Solution: dy y = dx + x ln y = arctan(x) + C y = Ke arctan(x) = Ke y = e arctan(x) Page of

3 MTH 33 Solutions to Final Exam May, 8 3. Determine whether each of the series converges or diverges. Show your work and justify your use of any convergence tests. 3 (a) (6 points) n n + n= Solution: Consider comparing with the series n. lim n 3 n n+ n 3n = lim n n n + = 3 Therefore series converges by the limit comparison test and since test. n= converges by the p-series n (b) (6 points) n= n n 3 (n + )! Solution: Consider the ratio test a n+ lim n a n So the series converges by the Ratio test = lim n n+ (n+) 3 (n+)! n n 3 (n+)! n+ (n + ) 3 = lim n (n + )! (n + )! n n 3 (n + ) 3 = lim n (n + ) n = < 3 Page 3 of

4 MTH 33 Solutions to Final Exam May, 8 4. (6 points) Evaluate the definite integral xe x dx. Solution: Consider integration by parts with u = x, dv = e x dx, du = dx and v = e x. xe x dx = [ x e x ] e x dx = [e] (e e ) = 5. (6 points) Find the Maclaurin series of the function f(x) = cos(x 4 ). Solution: cos(x 4 ) = = ( ) n (x4 ) n (n)! ( ) n x8n (n)! n= n= Page 4 of

5 MTH 33 Solutions to Final Exam May, 8 6. (a) (3 points) Shown on the plot below are the two polar curves r = 3 cos θ and r = + cos θ. Label the plot indicating which curve is which. 3 cos θ + cos θ (b) (5 points) Find all points of intersections between the two curves in part (a). Express your answer in polar coordinates (r, θ). Solution: From the picture we can see they intersect at the origin, that is, (r, θ) = (, ) is an intersection point where can be any value. Also the curves intersect at 3 cos θ = + cos θ cos θ = / θ = π/3, 5π/3 Which is at the points (r, θ) = (3/, π/3) and (r, θ) = (3/, 5π/3). (c) (4 points) Set up, but do not evaluate, an integral which computes the area in the first quadrant that is inside r = 3 cos θ and outside r = + cos θ. Solution: A = π/3 [ (3 cos θ) ( + cos θ) ] dθ Page 5 of

6 MTH 33 Solutions to Final Exam May, 8 7. Compute the following indefinite integrals. 4x + 3 (a) (6 points) x 3 + x dx Solution: 4x + 3 x 3 + x = 4x + 3 x(x + ) = A x + Bx + C x + which yields Ax + A + Bx + Cx = 4x + 3 giving A = 3, B = and C = 4x x 3 + x dx = x + x x + dx = ln x + ln(x + ) + C (b) (6 points) x 4 x dx Solution: Use trigonometric substitution with x = sin θ and dx = cos θ dθ. x dx = 4 sin θ ( cos θ dθ) 4 x 4 4 sin θ = 4 sin θ dθ = cos(θ)) dθ = θ sin(θ) ( x ) = sin x 4 x + C Page 6 of

7 MTH 33 Solutions to Final Exam May, 8 Multiple Choice. Circle the best answer. No work needed. No partial credit available. No credit will be given for choices not clearly marked. 8. (3 points) Let R be the region in the first quadrant bounded by the x-axis, the line x = 4, and y = x. The volume of the solid of revolution formed by revolving R about the x-axis is given by: A. B. C. D. E π y dy π y 4 dy π x dx π x dx π x dx 9. (3 points) The derivative of y = (ln x)(sin x) is equal to A. sin x x B. sin x x C. D. x x ln x x + ln x + x + ln x x E. None of the above.. (3 points) Evaluate lim x cos x x. A. B. / C. D. E. The limit does not exist. Page 7 of

8 MTH 33 Solutions to Final Exam May, 8. (3 points) The derivative of the function y = x x is equal to A. x x x B. x x ln x C. (x ) x D. x x ( + ln x) E. x x. (3 points) Which of the following series converge? (a) n= n (b) n= ( ) n n /3 (c) n= ( ) n n (d) n= n 3/ A. (b) and (c) only. B. (a), (b), and (d) only. C. (b) and (d) only. D. (b), (c) and (d) only. E. None of the above choices are correct. 3. (3 points) Which of the following series converge absolutely? (a) ( ) n n= n (b) n= n (c) ( ) n n= n (d) n= ( ) n n / A. (b) and (c) only. B. (a), (b), and (d) only. C. (b) and (d) only. D. (b), (c) and (d) only. E. None of the above choices are correct. Page 8 of

9 MTH 33 Solutions to Final Exam May, 8 4. (3 points) Evaluate lim ( + ) x. x + x A. B. /e C. D. e E. The limit does not exist. 5. (3 points) The Maclaurin series of the function + x starts with the terms: A. + x 4! x4 B. + x 8 x C. x + x + 3 x3 D. + x + 8 x E. x 3! x3 + 5! x5 6. (3 points) The radius of convergence of the power series A. 3 B. / C. D. E. 5 n= (x + 3) n n n is: Page 9 of

10 MTH 33 Solutions to Final Exam May, 8 7. (3 points) Let f(x) = x + x. Find (f ) () given that f() =. A. /3 B. 3/ C. + /( ) D. /( + ) E (3 points) The length of the curve given in polar coordinates by r = + cos θ, for θ π, is given by: π A. + sin θ dθ B. C. D. E. π π π π + cos θ dθ ( + cos θ) dθ + cos θ dθ + cos θ dθ 9. (3 points) Which of the following improper integrals converge? (a) x dx (b) x dx (c) x dx (d) x dx. A. (b) and (c) only. B. (a), (b), and (d) only. C. (b) and (d) only. D. (b), (c) and (d) only. E. None of the above choices are correct. Page of

11 MTH 33 Solutions to Final Exam May, 8 More Challenging Questions. Show all work to receive credit. Please BOX your final answer.. (6 points) Approximate the equation in x and y for the line tangent to the parametric curve (x(t), y(t)) at t = 6. The graphs of x(t) and y(t) are shown below. x(t) y(t) t t Solution: When t = 6 it appears x(6) = 3, x (6) =, y(6) = and y (6) =. In which case the tangent line would be y = (x 3). (6 points) Find all values of x for which the series cos n x diverges. (Make sure you justify both that the series diverges for the values of x in your list, and that the series converges for all other values.) n= Solution: A geometric series r n converges when r < and r = cos x has the range of [, ] so n= the only values on which the series diverges is when cos(x) = ± x = kπ (for all integers k) Page of

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