Problem Possible Points Points Earned Problem Possible Points Points Earned Test Total 100

Transcription

1 MATH 1080 Test 2-Version A Fall 2015 Student s Printed Name: Instructor: Section # : You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or any technology on any portion of this test. All devices must be turned off while you are in the testing room. During this test, any communication with any person (other than the instructor or the designated proctor) in any form, including written, signed, verbal, or digital, is understood to be a violation of academic integrity. No part of this test may be removed from the examination room. GENERAL DIRECTIONS: In order to receive full credit for the problems on the test, you must: 1. Read carefully and follow the directions given for each problem. 2. Show legible, organized, logical, and relevant justification which supports your final answer. 3. Use complete and correct mathematical notation. 4. Include proper units, if necessary. 5. All limits of integration and evaluations of definite integrals should be expressed as exact values and not as decimal approximations (i.e. π and not 3.14). 6. For an improper integral, state whether it converges or diverges. If it converges, give the value to which it converges. 7. Each part/problem should have a concluding statement, either a mathematical equation or a proper English sentence that provides the solution/answer and clearly identifies it as the solution/answer. Complete the information at the top of the page and sign your name below. You have 90 minutes to complete the entire test. On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test. Student s Signature: Do not write below this line. Problem Possible Points Points Earned Problem Possible Points Points Earned Test Total 100

2 1. Consider the curve defined by the parametric equations x = 1 3 cos(2θ) θ π 2. y = 3 sin(2θ) a. (6 points) Eliminate the parameter to find the second-degree Cartesian equation (a conic) of the curve. Write the equation in standard form and identify the type of conic. b. (5 points) Sketch the curve. Make sure to indicate direction and to label the center and/or vertices as appropriate.

3 2. (10 points) Find the area (if the area is finite) between the curve f(x) = and the x-axis for 0 x π. 1 cos 2 x

4 3. Consider the curve defined by the parametric equations x = cos(2t) 0 t π 3 y = sin (2t) Without eliminating the parameter and converting to a Cartesian equation, use the parametric equations and calculus to a. (6 points) find the exact maximum height of the curve on the given interval. b. (8 points) find the exact area between the curve and the x-axis on the given interval. Hint: Consider the graph of the curve. c. (7 points) find the exact length of the curve on the given interval.

5 4. Set up, but do not evaluate or simplify, the integral(s) that gives the surface area for the surface obtained by rotating a. (5 points) y = 2 + 5e 2x, 0 x 2 about the x-axis. b. (5 points) x = 5 ln(y) + y, 2 y e 2 about the x-axis. c. (5 points) y = arcsin(2x), 0 x 1 about the y-axis. 3 d. (5 points) x = y 4 y, 0 y 1 about the y-axis.

6 5. Consider the equation 9x 2 4y 2 36x = 0. a. (6 points) Find the standard form of the conic represented by this equation and identify the conic. b. (5 points) Sketch the conic from part a. and label the center and/or vertices as appropriate.

7 6. Consider the polar equations r 1 = 3 sin θ and r 2 = cos θ.. a. (5 points) Sketch the curves on the same axis provided below. b. (5 points) Using polar coordinates, set up but do not evaluate or simplify, the integral(s) that gives the area that lies inside both curves. c. (5 points) Using polar coordinates, set up but do not evaluate or simplify, the integral(s) that gives the area that lies inside r 2 and outside of r 1.

8 7. Determine whether the sequence converges or diverges. If it converges, find the limit. a. (6 points) a n = ln(5n 2 + 2) ln(4n) b. (6 points) { 3 cos2 n 5 n }