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:. 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. 5. Read the directions concerning infinite series at the beginning of the free response section. 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). 4(b). 3 (b). 9 5(a). 6 2. 4 5(b). 5 3. 0 Free Response 70 4(a). 2 Multiple Choice 30 Test Total 00 - Page of 4
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Multiple Choice: There are 0 multiple choice questions. They all have the same point value. Each question has one correct answer. The multiple choice problems will count for 30% of the total grade. Use a number 2 pencil and bubble in the letter of your response on the scantron sheet for problems - 0. 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. You are NOT permitted to use a calculator on any portion of this test.. (3 pts.) The Maclaurin series for f(x) = ln( + x) is ln( + x) = ( ) n+ x n, for < x. n Use the Maclaurin series for ln( + x) to determine the Maclaurin series for x 3 ln( + x 2 ). (a) ( ) n+ x 2n n (c) ( ) n+ x 3n+2 n (b) ( ) n+ x n+3 n (d) ( ) n+ x 2n+3 n 2. (3 pts.) Determine the coefficient of (x ) 2 in the Taylor series for f(x) = arctan x centered at a =. (a) 8 (b) 2 (c) 4 (d) 2 - Page 3 of 4
3. (3 pts.) Which of the following is the graph of the curve given by x 2 +9y 2 +4x+8y +4 = 0? (a) (c) (b) (d) 4. (3 pts) Determine the minimum number of terms of the convergent series must be summed so that the remainder is less than 9. ( ) n+ n 2 that (a) 4 (b) 5 (c) 2 (d) 3 - Page 4 of 4
5. (3 pts.) Given that the Maclaurin series for f(x) = e x is the following values does the series 2 n n! represent? x n, for < x <, which of n! (a) The series 2 n n! does not converge. (c) 2 n n! = e2 (b) 2 n n! = e 2 (d) 2 n n! = e 6. (3 pts.) The Maclaurin series for f(x) = x 2 sin(x 2 ) is x 2 sin(x 2 ) = Use the Maclaurin series for x 2 sin(x 2 ) to evaluate ( ) n x 4n+4, for < x <. (2n + )! x 2 sin(x 2 ) dx as a power series. (a) C + ( ) n+ x 4n+5 (2n + )!(n + )(4n + 5) ( ) n (4n + 4)x 4n+3 (c) C + (2n + )! (b) C + ( ) n x 4n+5 (2n + )!(4n + 5) (d) C + ( ) n (4n + 4)x 4n+3 (2n + )!(4n + 3) - Page 5 of 4
7. (3 pts.) Consider the series find that: (a) The series diverges. ( ) 5n 2n. By applying the Root Test to this series, we + 4n (c) The series converges absolutely. (b) The series converges conditionally. (d) The root test is inconclusive. 8. (3 pts.) Which of the following statements is TRUE about the series (a) The series converges by the Alternating Series Test. cos(nπ) n /3? (b) The series diverges by the Divergence Test. (c) The series converges absolutely by the Comparison Test. (d) The series diverges by the Integral Test. - Page 6 of 4
n + 3 9. (3 pts.) Consider the series 2n 2 + 4. Which of the following should we choose as the comparison series so that the Limit Comparison Test can be used to determine the convergence or divergence of this series? (a) n 2 (c) n (b) n 3/2 (d) n 5/2 0. (3 pts.) Suppose that 0 a n b n. Which of the following statements is TRUE? (a) If b n diverges, then a n diverges. (b) If a n converges, then b n converges. (c) If a n converges, then b n diverges. (d) If a n diverges, then b n diverges. - Page 7 of 4
Free Response. The Free Response questions will count for 70% 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. Give answers as exact values. You are NOT permitted to use a calculator or any other technology on any portion of this test. In addition, you must: For problems involving analysis of convergence or divergence of an infinite series, include: (a) Identification of the test or definition used in the analysis; (b) Work to show that the test conditions have been met, or work to apply the definition; (c) Conclusion statement about convergence or divergence.. Consider the power series (x + ) n 4 n+ n. (a) ( pts.) Use a test from the Series Summary Sheet to determine the radius of convergence, R, for the power series. - Page 8 of 4
(b) (9 pts.) Determine the interval of convergence for the power series Hint: Don t forget to check for convergence at the endpoints. (x + ) n 4 n+ n. - Page 9 of 4
2. (4 pts.) Using a test or tests from the Series Summary Sheet, determine whether the following series converges absolutely, converges conditionally, or diverges. ( ) n n n 2 + - Page 0 of 4
3. (0 pts.) Using a test or tests from the Series Summary Sheet, determine whether the following series converges or diverges. ( 2) n (n!) 2 - Page of 4
4. (a) (2 pts.) Find the 3rd-order Taylor polynomial, T 3 (x), of the function f(x) = cos x centered at a = π 6. (b) (3 pts.) Use Taylor s Inequality to estimate the accuracy of the approximation f(x) T 3 (x) when 0 x π 3. You do not need to simplify your answer. - Page 2 of 4
5. Consider the curve given by the following parametric equations: x = 3 cos(2t), y = 3 sin(2t), for π 2 t π. (a) (6 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 3 of 4
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 0 bubbled in answers; has MATH 080 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 4 of 4