Test 2 Version B. 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 1(a). 5 3(c). 5 1(b) (c). 5 5(a) (b). 10 3(a). 6 Free Response 70 3(b). 6 Multiple Choice 30 Test Total Page 1 of 12

2 Multiple Choice: There are 10 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 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.) Which one of the following sequences is both bounded and monotonic? { (a) 1, 1 2, 1 4, 1 8, 1 }, 16,... where a n = ( 1)n 1 2 n 1 (b) { 4 3, 16 9, 64 27, , 1024 }, 243,... where a n = 4n 3 n (c) { 1 2, 3 4, 7 8, 15 16, 31 } 32,..., where a n = n (d) {1, 2, 3, 4, 5,...}, where a n = ( 1) n+1 n 2. (3 pts.) Does the sequence whose nth term is a n = ln(6n 3 + 1) ln(3n n) converge? If so, to what value? (a) Yes, it converges to ln 2. (c) Yes, it converges to 0. (b) Yes, it converges to ln 3. (d) No, it diverges. - Page 2 of 12

3 3. (3 pts.) Does the sequence whose nth term is a n = en + 3n 1 + n (a) Yes, it converges to 3. converge? If so, to what value? (c) Yes, it converges to 1. (b) Yes, it converges to 0. (d) No, it diverges. 4. (3 pts.) Choose the correct partial fraction decomposition general form for the rational function 5x 2 + 2x 7 (x + 3)(x 2 9)(4x 2 + 1) 2. (a) A x B (x + 3) 2 + C x 3 + Dx + E 4x F x + G (4x 2 + 1) 2 (b) A x Bx + C x Cx + D (4x 2 + 1) 2 (c) A x B x 3 + C (x 3) 2 + D 4x E (4x 2 + 1) 2 (d) A x B (x 3) 2 + Cx + D 4x Ex + F (4x 2 + 1) 2 - Page 3 of 12

4 5. (3 pts.) Find the exact length of the curve y = 4 3 x3/2 on the interval [0, 2]. (a) 13 4 (c) 39 4 (b) 13 3 (d) (3 pts.) Consider the curve defined by x = y 2, for 3 y 4. Which of the following represents the arc length of the curve on the given interval? (a) 4 3 2y 3 2y 4 dy (c) 4 3 4y 7 4y 8 dy (b) 4 3 4y 9 4y 8 dy (d) 4 3 2y 5 2y 4 dy - Page 4 of 12

5 7. (3 pts.) Which one of the following integrals is NOT improper? (a) x 2 + 2x 3 dx (c) dx (x 3) 2/3 (b) π/2 0 sec 2 x dx (d) π/4 0 cot x dx 8. (3 pts.) Does the sequence whose nth term is a n = cos(nπ/2) n converge? If so, to what value? (a) No, it diverges. (c) Yes, it converges to 1. (b) Yes, it converges to 1. (d) Yes, it converges to 0. - Page 5 of 12

6 9. (3 pts.) Determine which one of the following series DIVERGES: (a) n=1 2 n 3 n (c) n=1 n 2 n 4 (b) n n n=1 (d) ( 3) n 8 n n=1 10. (3 pts.) For which one of the following integrals is integration by parts the best technique to use? ln x (a) x sin(x 2 ) dx (c) x 2 dx (b) e x x dx (d) arctan x 1 + x 2 dx - Page 6 of 12

7 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. 1. (15 pts.) (a) (5 pts.) Consider the curve defined by y = ln(1 + x 3 ) for 0 x 2. Completely set up but do not evaluate or simplify an integral with respect to x representing the area of the surface obtained by rotating the curve on the given interval about the x-axis. (b) (5 pts.) Consider the curve defined by y = cos(2x), for 0 x π/6. Completely set up but do not evaluate or simplify an integral with respect to x representing the area of the surface obtained by rotating the curve on the given interval about the y-axis. (c) (5 pts.) Consider the curve defined by x = 9 y 2 for 1 y 2. Completely set up but do not evaluate or simplify an integral with respect to y representing the area of the surface obtained by rotating the curve on the given interval about the y-axis. - Page 7 of 12

8 2. (14 pts.) Evaluate 8x 13 (2x + 1)(x 2 + 4) dx. - Page 8 of 12

9 3. For each of the following, determine whether the series converges or diverges. If the series converges, find its sum. Include the name of the convergence test or the definition used; work to show that the test conditions have been met or work to apply the definition; and a conclusion statement about convergence or divergence. (a) (6 pts.) k=1 ( ) 1 1 k + 2 k + 1 (b) (6 pts.) k=1 4 2 k 3 k+1 - Page 9 of 12

10 (c) (5 pts.) k=1 k 6 + 4k 1 + 2k 6 4. (10 pts.) Determine whether the following integral is convergent or divergent. Evaluate the integral if it is convergent. 7 4 dx (x 4) 2 - Page 10 of 12

11 5. (14 pts.) Consider the infinite series: ne 2n2. n=1 (a) (4 pts.) Show the conditions are met for the Integral Test. (b) (10 pts.) Use the Integral Test to determine if the series converges or diverges. - Page 11 of 12

12 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 10 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. 2 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 12 of 12

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