RED. Math 113 (Calculus II) Final Exam Form A Fall Name: Student ID: Section: Instructor: Instructions:

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1 Name: Student ID: Section: Instructor: Math 3 (Calculus II) Final Exam Form A Fall 22 RED Instructions: For questions which require a written answer, show all your work. Full credit will be given only if the necessary work is shown justifying your answer. Simplify your answers. Calculators are not allowed. Should you have need for more space than is allocated to answer a question, use the back of the page the problem is on and indicate this fact. Please do not talk about the test with other students until after the last day to take the exam. For Instructor use only. # Possible Earned MC # Possible Earned Sub 69 Sub 3 Total

2 Fall 22 Math 3 Final Exam Page 2 of 9 Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.. Which of the following series converge absolutely? ( ) n ) 2) n 2 ( ) n n 3) n 3 a) None b) c) 2 d) 3 e), 2 f), 3 g) 2, 3 h), 2, 3 2. Find the radius of convergence of the series n!(x 2) n n 2 3 n. a) b) c) 2 d) 3 e) 2/3 f) 4/3 g) /3 h) 2 i) 3 j) 3. What is the coefficient of x in the Maclaurin series of e 3x2? a) b) 3 c) d) 3! g) 3 5 5! e) ! h) 35 5! f) i) Diverges 4. When a particle is located a distance x feet from the origin, a force of x 2 + 2x pounds acts on it. How much work (measured in foot-pounds) is done in moving it from x = to x = 3? a) 6 b) 9 c) 5/3 d) 7/3 e) 6/3 f) 45/2 g) 27/2 h) 5/2 i) 5. The region between the curve y = and the x-axis for < x is rotated about the x-axis to xp form a solid of revolution. For which positive values of p does this solid have finite volume? a) < p b) < p < c) < p d) < p < /2 e) /2 < p f) < p < 2 g) 2 < p h) The volume is infinite for all positive p. 2

3 Fall 22 Math 3 Final Exam Page 3 of 9 6. Evaluate the integral π/2 sin 3 (x) cos 2 (x) dx. a) /5 b) 2/5 c) 3/2 d) /3 e) 3/5 f) π g) π/3 h) π/2 i) 7. Evaluate the integral dx + 4x 2. a) b) c) arcsin(/2) d) π e) 2π f) π/2 g) arctan(π) h) 2 i) 2 8. Which integral represents the length of the curve y = sin x + cos x, x π/4? (You might need to set up an integral and do a short calculation.) a) c) e) π/4 π/4 π/4 π/ cos x sin x dx b) 2 2 cos x sin x dx π/4 2 cos x sin x dx d) 2 cos x sin x dx π/4 cos x sin x dx f) + cos x sin x dx 9. Which integral represents the area of the surface obtained by rotating the curve about the y-axis. y = e x, y 8 a) c) e) 8 ln 8 8 2πx + e 2x dx b) 2πx + e 2x dx d) 2πe x + e x dx f) ln 8 ln 8 8 2π + e 2x dx 2πe x + e 2x dx 2πe x + e 2x dx. If ( x, ȳ) is the centroid of the region bounded by the line y = x and the parabola y = x 2, what is ȳ? a) b) /2 c) /3 d) 2/3 e) /4 f) 3/4 g) /5 h) 2/5 i) 3/5 3

4 Fall 22 Math 3 Final Exam Page 4 of 9. A curve is parametrized by the equations x = 6 sin t and y = t 2 + t. Find the slope of the line that is tangent to this curve at the point (, ). a) b) c) /2 d) 2 e) /3 f) 3 g) /6 h) 6 i) Undefined 2. Determine the exact value of the geometric alternating series: a) /2 b) 3/4 c) 7/6 d) 7/4 e) 3/8 f) 7/8 3. Which of the following three tests will establish that the series 3 n(n + 2) converges? ) Comparison Test with 2) Limit Comparison Test with 3) Comparison Test with 2n 2 3n 2 n 2 a) None b) c) 2 d) 3 e), 2 f), 3 g) 2, 3 h), 2, 3 4

5 Fall 22 Math 3 Final Exam Page 5 of 9 Part II: Written Response Neatly write the solution to each problem. Complete explanations are required for full credit. 4. (6 points) Find the volume of the solid obtained by rotating the region bounded by y = x x 2 and y = about the vertical line x = (6 points) Evaluate x sin(3x) dx. 5

6 Fall 22 Math 3 Final Exam Page 6 of 9 6. (6 points) Evaluate 4x2 dx. 7. (6 points) Evaluate the integral dx x 2 + x. 6

7 Fall 22 Math 3 Final Exam Page 7 of 9 8. (6 points) Let s(n) = n k= why this choice of n is large enough. k. Find a large enough value of n such that s(n) 2, and justify Hint: Think about the geometric reasoning used in the proof of the Integral Test. 9. (6 points) Determine the interval of convergence for the power series (2x ) n 5 n n. 7

8 Fall 22 Math 3 Final Exam Page 8 of 9 2. (6 points) Assuming < x <, evaluate the definite integral the answer using summation notation. x du as a power series. Express + u7 2. (7 points) Find the Taylor series for the function f(x) = x centered at the value a =. Express the answer using summation notation. 8

9 Fall 22 Math 3 Final Exam Page 9 of (6 points) Find the area of the region that lies inside the first curve and outside the second curve: r = 3 cos θ, r = + cos θ. 23. (6 points) A trough is full of water. Its end is shaped like the shaded region in the picture. The boundaries of the region are the curves y = and y = + x /2 for x. If the pressure at depth d is P = δd, where δ is a constant and d is measured in meters, set up a definite integral for the hydrostatic force F against the end of the trough. [Note: Set up an integral for F, but don t evaluate the integral. The answer will involve δ.] y x y x END OF EXAM 9

Part I: Multiple Choice Mark the correct answer on the bubble sheet provided. n=1. a) None b) 1 c) 2 d) 3 e) 1, 2 f) 1, 3 g) 2, 3 h) 1, 2, 3

Math (Calculus II) Final Eam Form A Fall 22 RED KEY Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.. Which of the following series converge absolutely? ) ( ) n 2) n 2 n (