Test 2 - Answer Key Version A

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1 MATH 8 Student s Printed Name: Instructor: CUID: Section: Fall 27 8., 8.2,. -.4 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. You have 9 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. 4 5(a). 8 2(a). 8 5(b). 4 2(b). 8 5(c). 4 3(a). 4 Free Response 7 3(b). Multiple Choice 3 4. Test Total - Page of 6

2 MATH 8 Fall 27 8., 8.2,. -.4 Multiple Choice: There are multiple choice questions. They all have the same point value. Each question has one correct answer. The multiple choice problems will count for 3% 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.. (3 pts.) Does the sequence whose nth term is a n = + ( )n n (a) Yes, it converges to -. (c) Yes, it converges to. converge? If so, to what value? (b) Yes, it converges to. (d) No, it diverges. Answer: (b) 2. (3 pts.) Which of the following integrals is NOT improper? (a) 3 (x ) 2 dx (c) π/2 tan x dx (b) x x 2 dx (d) π/4 π/4 sec x dx Answer: (d) - Page 2 of 6

3 MATH 8 Fall 27 8., 8.2, (3 pts.) Consider the series k= ( 2 k + 2 ). Determine the partial sum S n for this series. k (a) S n = 2 n (c) S n = + 2 n + (b) S n = n + (d) S n = 2 2 n Answer: (b) ( ) 2n 4. (3 pts.) Does the sequence whose nth term is a n = arctan converge? If so, to what + 2n value? (a) Yes, it converges to π 4. (c) No, it diverges. (b) Yes, it converges to. (d) Yes, it converges to π 2. Answer: (a) - Page 3 of 6

4 MATH 8 Fall 27 8., 8.2, (3 pts.) Is the series convergent or divergent? If it is convergent, what is the sum? 3n+ n= (a) convergent; sum = 7 2 (c) convergent; sum = 7 6 (b) divergent (d) convergent; sum = 7 3 Answer: (c) 6. (3 pts.) Choose the correct partial fraction decomposition general form for the rational function x 3 + x (x 2 + 7x + 2)(x 2 + x + ) 2. (a) A x B x Cx + D x 2 + x + + Ex + F (x 2 + x + ) 2 (b) A x B x C x + + D (x + ) 2 + E (x + ) 3 + F (x + ) 4 (c) Ax + B x 2 + 7x Cx + D (x 2 + x + ) 2 (d) A x Answer: (a) B x C x 2 + x + + D (x 2 + x + ) 2 - Page 4 of 6

5 MATH 8 Fall 27 8., 8.2, (3 pts.) Consider the curve defined by y = x + for x 3. Which of the following gives the area of the surface obtained by rotating the curve on the given interval about the x-axis? (a) 3 4x + 5 2πx 4x + 4 dx (c) 3 x + 2 2πx x + dx (b) 3 2π 4x + 5 x + 4x + 4 dx (d) 3 2π x + x + 2 dx Answer: (b) 8. (3 pts.) For which of the following integrals is integration by parts the best technique to use? ln x (a) x dx (c) x 2 e 3x dx (b) 2xe x2 dx (d) x x 2 + dx Answer: (c) - Page 5 of 6

6 MATH 8 Fall 27 8., 8.2, (3 pts.) Does the sequence whose nth term is a n = 4 sin n n 3 (a) Yes, it converges to 4. (c) No, it diverges. converge? If so, to what value? (b) Yes, it converges to. (d) Yes, it converges to. Answer: (b). (3 pts.) Using the Comparison Test, the series (a) Diverges because k= k 3/2 + : k 3/2 + < k 3/2 and diverges. k3/2 k= (b) Converges because k 3/2 + > k 3/2 and converges. k3/2 k= (c) Diverges because k 3/2 + > k 3/2 and diverges. k3/2 k= (d) Converges because Answer: (d) k 3/2 + < k 3/2 and converges. k3/2 k= - Page 6 of 6

7 MATH 8 Fall 27 8., 8.2,. -.4 Free Response. The Free Response questions will count for 7% 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. Note on notation deductions: deduct.5 points for missing more than one equal sign on a problem; deduct.5 points for missing more than one dx on a problem.. (4 pts.) Evaluate 5x 2 4x + 8 x 2 (x 2 + 4) dx. Solution: We first rewrite the integrand using partial fraction decomposition. 5x 2 4x + 8 x 2 (x 2 + 4) = A x + B x 2 + Cx + D x So Equating the coefficients we have, 5x 2 4x + 8 = Ax(x 2 + 4) + B(x 2 + 4) + (Cx + D)x 2 () = Ax 3 + 4Ax + Bx 2 + 4B + Cx 3 + Dx 2 = (A + C)x 3 + (B + D)x 2 + 4Ax + 4B x 3 : = A + C x 2 : 5 = B + D x : 4 = 4A x : 8 = 4B Solving this system, we find A =, B = 2, C =, D = 3. So 5x 2 4x + 8 x 2 (x 2 + 4) ( dx = x + 2 x 2 + x + 3 ) x 2 dx = x dx + x 2 dx + x x dx + = ln x + 2 x + x x dx + 3 x dx 3 4((x/2) 2 + ) dx We use substitution on the two integrals. Let u = x 2 + 4, then du = 2x dx = du = x dx. 2 Let w = x 2, then dw = 2 dx = 2dw = dx. So ln x + 2 x + x x dx + 3 4((x/2) 2 + ) 2 dx = ln x + x + 2 u du w 2 + dw = ln x + 2 x + 2 ln u arctan w + C = ln x + 2 x + 2 ln(x2 + 4) + 3 ( x ) 2 arctan + C 2 - Page 7 of 6

8 MATH 8 Fall 27 8., 8.2,. -.4 So 5x 2 4x + 8 x 2 (x 2 + 4) dx = ln x + 2 x + 2 ln(x2 + 4) + 3 ( x ) 2 arctan + C 2 Work on Problem: Determines the correct form of the PFD Clears fractions to get () Determines A, B, C, D with correct supporting work Rewrites the integral as an integral of the PFD Integrates /x Integrates 2/x 2 Integrates x/(x 2 + 4) Integrates 3/(x 2 + 4) Notes: -.5 point for notation errors (max deduction of for notation errors) -.5 point for missing +C -.5 for wrong sign or copying error -.5 point for missing absolute value Points 3 points 3 points - Page 8 of 6

9 MATH 8 Fall 27 8., 8.2, Determine whether each of the following series converges or diverges. 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. (a) (8 pts.) n= 2n 5 + n Solution: We observe that So by the Divergence Test, the series Work on Problem: Considers lim n a n Evaluates the limit Notes the limit is not lim n n= 2n 5 + n = 2 2n 5 + n diverges. Concludes series diverges by the Divergence Test Notes: Deduct.5 points for each notation error. (with a max. deduction of for notation errors) n (b) (8 pts.) (n 2 + ) 2 n= n= Points 3 points Solution: Since the terms of the series are positive, we try a comparison test. n Note that (n 2 + ) 2 = n n 4 + 2n 2 +. n= We try the limit comparison test and compare the given series to by the p-series test since p = 3 >. So the series n= lim n n n 4 + 2n 2 + n 3 n 4 n n 4 + 2n 2 + n + 2/n 2 + /n 4 = > n (n 2 converges by the limit comparison test. + ) 2 n=, which converges n3 - Page 9 of 6

10 MATH 8 Fall 27 8., 8.2,. -.4 Work on Problem: States an appropriate comparison series States the comparison series converges with justification Sets up the limit for the limit comparison test Simplifies the limit Evaluates the limit Observes that the limit is finite and > Concludes that the series converges by the limit comparison test Notes: Deduct.5 points for algebra or notation errors. Points - Page of 6

11 MATH 8 Fall 27 8., 8.2, (4 pts.) Consider the infinite series: n= 3 2n +. (a) (4 pts.) Show the conditions are met for the Integral Test. Solution: Let f(x) = (2x + ) /3. Then () f(x) is positive for x and (2) f(x) is continuous for x. f (x) = 2 3(2x + ) 4/3. So f (x) < for all x /2, so f(x) is 3 (2x + ) 4/3 (2) = decreasing for all x /2, and therefore (3) f(x) is decreasing for x. So the conditions are met for the integral test. (b) ( pts.) Use the Integral Test to determine if the series converges or diverges. Solution: (2x + ) /3 dx b b (2x + ) /3 dx We use substitution. Let u = 2x +, so du = 2dx 2du = dx. Then lim b b (2x + ) /3 dx b 2 b 2 2b+ 3 ( 3 2 u2/3 u /3 du ) 2b+ 3 b 4 ((2b + )2/3 3 2/3 ) = 3 Since the integral integral test. (2x + ) /3 dx diverges, the series n= 3 2n + diverges by the Work on Problem: Work in terms of x States f(x) is positive and continuous for x States f(x) is decreasing for x (with justification) States the appropriate improper integral Re-writes using a limit Finds the antiderivative Plugs in the limits of integration Evaluates the limit Concludes series diverges because integral diverges Notes: Deduct.5 points for each notation error. (with a max. deduction of for notation errors) Points 3 points - Page of 6

12 MATH 8 Fall 27 8., 8.2, ( pts.) Determine whether the following integral is convergent or divergent. Evaluate the integral if it is convergent. x ln x dx Solution: x ln x dx a + a x ln x dx We use integration by parts. Let u = ln x and dv = xdx, so du = x2 xdx and v = lim a + a x ln x dx a + a + a + a + [ a2 a + a + ] x dx ] [ x 2 2 ln x x 2 a a 2 [ x 2 2 ln x x dx a 2 a [ x 2 ] 2 ln x x2 a 4 a ( )] [ a2 2 ln a 4 a2 4 2 ln a ] 4 + a2 4 [ ln a 2a 2 ] 4 + a So a 2 where lim a + 4 = and lim a + ln a 2a 2 is an indeterminate form. By L Hospital s Rule, lim ln a a + 2a 2 /a a + 4a 3 a 2 a + 4 = So So x ln x dx converges. lim a + a x ln x dx = 4 - Page 2 of 6

13 MATH 8 Fall 27 8., 8.2,. -.4 Work on Problem: Re-writes the integral using a limit Re-writes integral with integration by parts work Finds the antiderivative Evaluates the integral (plugs in limits of integration) Evaluates the limit (with justification using L Hospital s Rule) Concludes that the improper integral converges Notes: Deduct.5 points for each notation error. (with a max. deduction of for notation errors) no more than 6 points total if improper integral not written as a limit at any point Points 4 points - Page 3 of 6

14 MATH 8 Fall 27 8., 8.2, (6 pts.) (a) (8 pts.) Find the arc length of the curve x = 3 (y2 + 2) 3/2 for y 2. Solution: Note that x = (y2 + 2) /2 (2y) = y y 2 + 2, so (x ) 2 = y 2 (y 2 + 2). So L = 4 3. L = Work on Problem: Finds x 2 Sets up the arc length integral correctly Expands + (x ) 2 and factors Simplifies the integrand Finds the antiderivative Evaluates the integral Notes: Deduct.5 points for each notation error 2 + y 2 (y 2 + 2) dy = + y 4 + 2y 2 dy = = 2 2 (y 2 + ) 2 dy (y 2 + ) dy ( ) y 3 2 = 3 + y = = 4 3 (with a max. deduction of for notation errors) Points 3 points.5 points.5 points - Page 4 of 6

15 MATH 8 Fall 27 8., 8.2,. -.4 (b) (4 pts.) Consider the curve defined by y = e 2x, x. 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. Solution: S = Work on Problem: Constant 2π Correct radius Correct ds component (with for y ) 2πx + (2e 2x ) 2 dx Points.5 points.5 points (c) (4 pts.) Consider the curve defined by x = tan y, y π/4. 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. Solution: S = π/4 Work on Problem: Constant 2π Correct radius Correct ds component (with for x ) 2π tan y + (sec 2 y) 2 dy Points.5 points.5 points - Page 5 of 6

16 MATH 8 Fall 27 8., 8.2,. -.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 bubbled in answers; has MATH 8 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 6 of 6

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