MATH 8 Student s Printed Name: Instructor: Test - Answer Key Spring 6 8. - 8.3,. -. 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, PDA, or any technology on either portion of this test. All devices must be turned o 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.. 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 Free Response Possible Problem Earned Problem Earned. 5.. 9 6. 3 3(a). Free Response 7 3(b). 4 Multiple Choice 3 4. Test Total - Page of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 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 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. You are NOT permitted to use a calculator on any portion of this test.. (3 pts.) If three equal subdivisions of [ 4, ] are used, what is the trapezoidal approximation of 4 e x? (a) 3 (e4 +4e +e + e ) (c) e 4 +e +e + e (b) (e4 +e +e + e ) (d) (e4 + e + e + e ) Answer: (b). (3 pts.) Choose the correct partial fraction decomposition general form for the rational function x + x + x(x +4x + 4)(x + 5). (a) A x + (b) A x + (c) A x + (d) A x + Bx + C x +4x +4 + Dx + E x +5 + Fx+ G (x + ) 5 B x + + Answer: (d) C (x + ) + E x +5 + Bx + C x +4x +4 + Cx + D (x + 5) F (x + 5) B x + + C (x + ) + Dx + E x +5 + Fx+ G (x + 5) - Page of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 3. (3 pts.) Which of the following represents the area of the surface obtained by rotating x =3y 4y for apple y apple about the x-axis? (a) y p 48y + 64y dy (c) (3y 4y ) p 48y + 64y dy (b) (3y 4y ) p 4 8y dy (d) y p 8 + 48y 64y dy Answer: (a) 4. (3 pts.) Does the sequence whose nth term is a n = tan n ln e + n converge? If so, to what value? (a) No, it diverges. (c) Yes, it converges to. (b) Yes, it converges to. (d) Yes, it converges to. Answer: (d) - Page 3 of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 5. (3 pts.) Which one of the following sequences diverges? n n o (a) n n ln n (b) (c) Answer: (a) e n e n (d) ( ) n+ n 6. (3 pts.) Which of the following statements are TRUE? I. If lim n! a n =, then II. If X a n = L, then n= III. The series X n= X a n converges. n= X a n = L + a. n= n (n + ) diverges. (a) I, II, and III (c) II only (b) I and III only (d) II and III only Answer: (d) - Page 4 of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 7. (3 pts.) Which one of the following is an improper integral? (a) +x (c) /3 sin x cos x (b) x x (d) p x + Answer: (b) 8. (3 pts.) Which one of the following improper integrals diverges? (a) +x (c) x /3 (b) e x + (d) x 3 + Answer: (c) - Page 5 of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 9. (3 pts.) Determine which of the following statements about sequences is FALSE. (a) If {a n } diverges and {b n } diverges, then {a n + b n } diverges. (b) If {a n } converges to and {b n } is bounded, then {a n b n } converges to. (c) If lim n! a n = L, then lim n! a n+3 = L. (d) If {a n } converges to 3 and {b n } converges to, then {a n + b n } converges to 5. Answer: (a). (3 pts.) Find the sum of the series X k= 5 k 5 k+. (a) The sum of the series is. (c) The sum of the series is 5. (b) The series diverges. Answer: (c) (d) The sum of the series is 5. - Page 6 of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 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. Give answers as exact values. You are NOT permitted to use a calculator or any other technology on any portion of this test.. ( pts.) Find the partial fraction decomposition of Show all work needed to determine your coe cients. 4x x x (x + x + 3). So 4x x x (x + x + 3) = A x + B x + Cx + D x + x +3 4x x = Ax(x + x + 3) + B(x + x + 3) + (Cx + D)x () =Ax 3 + Ax +3Ax +Bx + Bx +3B + Cx 3 + Dx =(A + C)x 3 +(A +B + D)x +(3A + B)x +3B Equating the coe cients we have, x 3 :=A + C x : x : x : 4=A +B + D =3A + B = 3B Solving this system we find: B = 4, A =, C =, and D = 3. So Determines the correct form of the PFD Clears fractions to get () 4x x x (x + x + 3) = x + 4 x + x +3 x + x +3 4 points point Determines the coe cients with correct supporting work 5 points Deduct.5 points for algebra or notation errors. - Page 7 of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -.. (9 pts.) Let f(x) be a rational function whose partial fraction decomposition is Evaluate f(x). 5x + + 3x 4x + f(x) = 5x + + 3x 4x + x = +3 5x + 4x + = 5 ln 5x + +3 x 4x + 4x + 4x + We will use substitution on the two integrals. Let u =4x +. Then du =8x,so du = x. 8 Also let w =x. Thendw =, so dw =. So ln 5x + +3 5 So x 4x + f(x) = 5 ln 5x + + 3 8 ln(4x + ) Integrates the first term Splits the second term into two integrals Integrates the remaining two integrals -.5 point for missing +C -.5 point for missing or more s -.5 for wrong sign or copying error 4x + = 5 ln 5x + + 3 du 8 u w + dw = 5 ln 5x + + 3 8 ln u arctan w + C = 5 ln 5x + + 3 8 ln(4x + ) arctan(x)+c -.5 point for wrong constant in front of each antiderivative -.5 point for having anything other than x in arctan(x) -.5 point for missing absolute value - for algebra error point.5 points each arctan(x)+c - Page 8 of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 3. (4 pts.) Consider the curve defined by y = (ex + e x ) on the interval apple x apple. (a) ( pts.) Find the arc length of the curve on the given interval. Note that y = (ex e x ). L = So L = (e e ). s Finds y + (ex e x ) = Sets up the arc length integral correctly Evaluates the integral Deduct.5 points for each notation error = = = r + 4 (ex +e x ) r e x 4 + + e x 4 s e x + e x e x + e x = (ex e x ) (with a max. deduction of point for notation errors) point 4 points 5 points = (e e ) (b) (4 pts.) Set up but do not evaluate an integral representing the area of the surface obtained by rotating the curve on the given interval about the x-axis. S = s (ex + e x ) + (ex e x ) Correct limits Constant Correct radius Uses arc length from set-up as found in (a) Deduct.5 points for each notation error. (with a max. deduction of point for notation errors) point point point point - Page 9 of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 4. ( pts.) Determine whether the following series converges or diverges. If it converges, find its sum. X i= i(i + ) So i(i + ) = A i + B i + =A(i + ) + Bi If we let i =, we find B =. If we let i =, we find A =. So X i= i(i + ) = X i i= i + We determine if this series converges using the definition of convergence of a series. So we need to consider {S n }, the sequence of partial sums and compute lim n! S n. nx S n = i + i + i= = + 3 4! + + n n =+ n + + n + + 3 5 4 6! + n + n n + n + So lim S n = lim + n! n! n + = 3 n + Since the sequence of partial sums converges to 3,theseries X Re-writes the series using partial fractions Considers S n Writes out the terms in S n Simplifies S n Finds lim n! S n Concludes that the series converges to 3/ Deduct.5 points for each notation error. (with a max. deduction of point for notation errors) i= point points points point i(i + ) converges to 3. - Page of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 5. ( pts.) Find the area (if the area is finite) of the region bounded by the graphs of y =ln y =, and x = for <xapple. x, A = ln x = lim a! + a ln x = lim a! + a ln(x ) = lim a! + a ln x We use integration by parts. Let u =lnx and dv =. Thendu = x and v = x. So lim a! + a ln x= lim x ln x a! + a = lim a! + a a ln a (x) a x x = lim ( a! + a ln a ( a)) = lim ( a! + a ln a +a) = lim a( a! + ln a) where the limit is I.F. = ln a lim a! + /a where the limit is I.F. = /a lim a! + /a by L Hospital s Rule = lim a! + = () = So the integral converges to. The area of the region is. Sets up the area integral Re-writes the integral using a limit Uses integration by parts Evaluates the integral (plugs in limits of integration) Finds the limit Deduct.5 points for each notation error. (with a max. deduction of point for notation errors) point points 4 points points - Page of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 6. (3 pts.) Find the centroid of the region bounded by the curves y = x /3 and y = x. A = (x /3 x) = 3x5/3 5! x = 3 5 = x = x(x /3 x) = = 3 = 8 (x 5/3 x ) 3x 8/3 8! x 3 3 = 5 3 So the centroid is ȳ = (x /3 ) x =5 5,. =5 3 =5 7 (x 4/3 x ) 3x 7/3 7! x 3 3 = 3 Finds the area Sets up the integral for x Evaluates the integral for x Sets up the integral from ȳ Evaluates the integral for ȳ States the coordinates of the centroid Deduct.5 points for each notation error. (with a max. deduction of point for notation errors).5 points.5 points point - Page of 3
MATH 8 Test - Answer Key Spring 6 8. - 8.3,. -. 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. 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 3 of 3