Test 3 Version A. On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test.
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1 Student s Printed Name: Instructor: CUID: Section: Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop, tablet, SMART watch, 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 a 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. Read each question carefully. In order to receive full credit for the free response portion of the test, you must: 1. Show legible and logical (relevant) justification that supports your final answer. 2. Use complete and correct mathematical notation. 3. Include proper units, if necessary. 4. Give exact numerical values whenever possible. 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. Free Response Possible Points Free Response Possible Points Problem Points Earned Problem Points Earned (a) (b) (a). 9 Free Response 69 4(b). 6 Multiple Choice 31 Test Total Page 1 of 14
2 Multiple Choice: There are 13 multiple choice questions. They do NOT all have the same point value. Each question has one correct answer. The multiple choice problems will count for 31% of the total grade. Use a number 2 pencil and bubble in the letter of your response on the scantron sheet for problems For your own record, also circle your choice on your test since the scantron will not be returned to you. Only the responses recorded on your scantron sheet will be graded. 1. (3 pts.) The Maclaurin series for f(x) = arctan x is arctan x = ( 1) n x 2n+1, for x 1. 2n + 1 Use the Maclaurin series for arctan x to determine the Maclaurin series for x 3 arctan(x 2 ). (a) ( 1) n x 2n+4 2n + 1 (c) ( 1) n x 4n+5 2n + 1 (b) ( 1) n x 4n+4 2n + 1 (d) ( 1) n x 4n+8 2n (1 pt.) Determine whether the series sin(2n) is absolutely convergent, conditionally con- 1 + n3 vergent, or divergent. (a) The series is absolutely convergent. (c) The series is divergent. (b) The series is conditionally convergent. - Page 2 of 14
3 3. (3 pts.) The Maclaurin series for f(x) = xe x3 is ( 1) n x 3n+1 xe x3 =, for < x <. n! Use the Maclaurin series for xe x3 to evaluate xe x3 dx as a power series. (a) C + ( 1) n+1 x 3n+2 (n + 1)!(3n + 2) ( 1) n (3n + 1)x 3n (c) C + n! ( 1) n (3n + 1)x 3n (b) C + n!(3n) (d) C + ( 1) n x 3n+2 n!(3n + 2) ( 1) n+1 4. (3 pts.) Consider the convergent series n 2. If the first five terms are used to estimate + 3 the value of the sum, then the bound on the absolute error given by the Alternating Series Estimation Theorem is: (a) 1 52 (c) 1 19 (b) 1 28 (d) Page 3 of 14
4 5. (1 pt.) Determine whether the series convergent, or divergent. (a) The series is absolutely convergent. ( 3 n ( 1) 4) n is absolutely convergent, conditionally (c) The series is divergent. (b) The series is conditionally convergent. 6. (3 pts.) If 0 x 2π/3, use Taylor s Inequality to find the best estimate of the maximum 3 error in the approximation sin x T 2 (x), where T 2 (x) = ( x π ) 3 ( x π ) 2 is the second-order Taylor polynomial of sin x at π/3. (a) π2 36 (c) π3 27 (b) π (d) π2 9 - Page 4 of 14
5 7. (3 pts.) Which of the following is the graph of the curve given by y 2 4x x + 2y = 39? (a) (c) (b) (d) - Page 5 of 14
6 8. (3 pts.) Consider the series series, we find that: ( 3n ( 1) n+1 2 ) n + 4n 2n 2. By applying the Root Test to this + 1 (a) The series converges absolutely. (c) The series diverges. (b) The series converges conditionally. (d) The Root Test is inconclusive. 9. (3 pts.) Find the power series representation for f (x) if f(x) = ( 1) n x 4n. (2n)! (a) f (x) = ( 1) n 2x 4n 1 (2n 1)! (c) f (x) = ( 1) n x 4n+1 (2n)!(4n + 1) (b) f (x) = ( 1) n+1 x 4n+1 (2n)!(4n + 1)(n + 1) (d) f ( 1) n 1 2nx 4n 1 (x) = (2n 1)! - Page 6 of 14
7 10. (1 pt.) Determine whether the series convergent, or divergent. ( 1) n+1 n 2 3n is absolutely convergent, conditionally (a) The series is absolutely convergent. (c) The series is divergent. (b) The series is conditionally convergent. 11. (3 pts.) Determine the third degree term in the Taylor series for f(x) = 1/x 2 centered at a = 1. (a) 4(x + 1) 3 (c) 6(x + 1) 3 (b) 24(x + 1) 3 (d) 8(x + 1) 3 - Page 7 of 14
8 12. (3 pts.) Using the Comparison Test, (a) Converges because 5 n 3 n 4 : 5 n 3 n 4 > 5n 3 n and 5 n 3 n converges. (b) Diverges because 5 n 3 n 4 > 5n 3 n and 5 n 3 n diverges. (c) Converges because 5 n 3 n 4 < 5n 3 n and 5 n 3 n converges. (d) Diverges because 5 n 3 n 4 < 5n 3 n and 5 n 3 n diverges. 13. (1 pt.) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. (a) The series is absolutely convergent. ( 1) n n 2/3 (c) The series is divergent. (b) The series is conditionally convergent. - Page 8 of 14
9 Free Response. The Free Response questions will count for 69% of the total grade. Read each question carefully. To receive full credit, you must show legible, logical, and relevant justification which supports your final answer. For problems involving analysis of convergence or divergence of series, remember to include: the name of the convergence test used, work to show that the test conditions have been met, and a conclusion statement about convergence or divergence of the series. 1. (11 pts.) Determine whether the following series converges or diverges: ( 1) n+1 n 2n 3/ Page 9 of 14
10 2. (11 pts.) Determine whether the following series converges or diverges: 4 n n! n 2 3. (11 pts.) Determine whether the following series converges or diverges: (n + 1) 2 4n Page 10 of 14
11 4. Consider the power series ( 1) n (x 5) n n3 n. (a) (9 pts.) Determine the radius of convergence, R, for the power series. (b) (6 pts.) Determine the interval of convergence for the power series. Hint: Don t forget to check for convergence at the endpoints. - Page 11 of 14
12 5. Consider the curve given by the following parametric equations: x = 2 sin t, y = 2 cos t, 0 t π/2. (a) (5 pts.) Eliminate the parameter to find a Cartesian equation of the curve. Be sure to include any necessary restrictions. (b) (5 pts.) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. Include the coordinates of the initial and terminal points with the appropriate value of t in the blanks below. Coordinates of initial point: at t = Coordinates of terminal point: at t = - Page 12 of 14
13 6. (11 pts.) Find the 3rd-order Taylor polynomial, T 3 (x), of the function f(x) = ln(x 2) centered at a = 3. - Page 13 of 14
14 Scantron: Check to make sure your Scantron form meets the following criteria: My Scantron: is bubbled with firm marks so that the form can be machine read; is not damaged and has no stray marks (the form can be machine read); has 13 bubbled in answers; has MATH 1080 and my Section number written at the top; has my Instructor s last name written at the top; has Test No. 3 written at the top; has the correct test version written at the top and bubbled in below my XID; shows my correct XID both written and bubbled in. **Bubble a zero for the leading C in your XID**. - Page 14 of 14
Test 3 - Answer Key Version B
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