Test 1 - Answer Key Version A

Transcription

1 MATH 8 Test - Answer Key Sring 6 Sections , Student s Printed Name: Instructor: CUID: Section: Instructions: You are not ermitted to use a calculator on any ortion of this test. You are not allowed to use any textbook, notes, cell hone, lato, PDA, or any technology on either ortion of this test. All devices must be turned o while you are in the testing room. During this test, any communication with any erson (other than the instructor or a designated roctor) in any form, including written, signed, verbal, or digital, is understood to be a violation of academic integrity. No art of this test may be removed from the examination room. Read each question carefully. In order to receive full credit for the free resonse ortion of the test, you must:. Show legible and logical (relevant) justification that suorts your final answer.. Use comlete and correct mathematical notation. 3. Include roer units, if necessary.. Give exact numerical values whenever ossible. You have 9 minutes to comlete the entire test. On my honor, I have neither given nor received inaroriate or unauthorized information at any time before or during this test. Student s Signature: Do not write below this line. Free Resonse Possible Points Points Earned Problem Free Resonse 7 Multile Choice 3 Test Total - Page of 5

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3 MATH 8 Test - Answer Key Sring 6 Sections , Multile Choice: There are 8 multile choice questions. They do not all have the same oint value. Each question has one correct answer. The multile choice roblems will count for 3% of the total grade. Use a number encil and bubble in the letter of your resonse on the scantron sheet for roblems - 8. For your own record, also circle your choice on your test since the scantron will not be returned to you. Only the resonses recorded on your scantron sheet will be graded. You are NOT ermitted to use a calculator on any ortion of this test.. ( ts.) Suose f is continuous and f and f have the values given below. Evaluate 3 xf (x) dx. x x x 3 f(x) 5 8 f (x) 3 (a) 9 (b) 5 (c) 5 (d) 3 Answer: (d). ( ts.) Evaluate the integral cos (3x) dx. (a) x + sin(6x)+c (c) x + 6 sin(3x)+c (b) 9 cos3 (3x)+C (d) x sin(6x)+c Answer: (a) - Page 3 of 5

4 MATH 8 Test - Answer Key Sring 6 Sections , ( ts.) When evaluating 9 5x and dx are relaced by 5x dx by trigonometric substitution, the exressions (a) 9 5x 3sin and dx 3 cos d 5 (c) 9 5x 3sin and dx 3 sin d 5 (b) 9 5x 3 cos and dx 3 sin d 5 (d) 9 5x 3 cos and dx 3 cos d 5 Answer: (d). ( ts.) To derive the formula for integration by arts, one would use: (a) The Mean Value Theorem (c) The Product Rule (b) The Chain Rule (d) None of these Answer: (c) - Page of 5

5 MATH 8 Test - Answer Key Sring 6 Sections , ( ts.) Which of the following integrals gives the volume of the solid whose base is bounded by the grahs of y x + and y x and whose cross sections erendicular to the base and arallel to the y-axis are isosceles triangles with height equal to twice the base? See figures below. (a) x + (x ) dx (c) x + (x ) dx (b) x + (x ) dx (d) x x + (x ) dx Answer: (c) 6. ( ts.) At what oint does the function f(x) 3x equal its average value on the interval [, ]? (a) (b) 3 (c) (d) Answer: (b) - Page 5 of 5

6 MATH 8 Test - Answer Key Sring 6 Sections , ( ts.) A force of ounds is required to stretch a sring.5 feet from its natural osition. Assuming Hooke s Law alies, how much work is done in stretching the sring 3 feet from its natural osition? (a) 5 ft-lb (b) 7 ft-lb (c) 36 ft-lb (d) 8 ft-lb Answer: (d) 8. ( ts.) Find the area of the region bounded by the curves y x 3 x and y x +x for ale x ale. (a) 3 (b) 7 (c) 9 (d) 55 Answer: (a) - Page 6 of 5

7 MATH 8 Test - Answer Key Sring 6 Sections , Free Resonse. The Free Resonse 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 suorts your final answer. Give answers as exact values. You are NOT ermitted to use a calculator or any other technology on any ortion of this test.. ( ts.) Evaluate the integral: x e 3x dx. Solution: Use integration by arts. Let u x and dv e 3x dx. Thendu xdxand v e 3x To integrate du dx and v e 3x x e 3x dx 3 x e 3x 3 x e 3x + 3 xe 3x 3 dx xe 3x dx 3.So xe 3x dx, use integration by arts again. Let u x and dv e 3x dx. Then 3.So 3 x e 3x + 3 xe 3x dx 3 x e 3x + ale 3x e 3x xe dx 3 x e 3x 9 xe 3x + e 3x dx 9 3 x e 3x 9 xe 3x 7 e 3x + C Therefore, x e 3x dx 3 x e 3x 9 xe 3x 7 e 3x + C. Work on Problem: Defines u, dv and finds du, v for first IBP Rewrites integral using integration by arts formula Defines u, dv and finds du, v for second IBP Rewrites integral using integration by arts formula Finishes integrating integrates R e 3x 3 dx Final answer with + C Notes: Deduct.5 oints for notation errors. Points oints ( oint each) oints oints ( oint each) oints oint oint - Page 7 of 5

8 MATH 8. ( ts.) Evaluate the integral: Test - Answer Key x x +x +8 dx. Sring 6 Sections , Solution: First comlete the square. x +x +8 (x + ) +. Then use a trig substitution. Let x + tan. Thendx sec d.so x x +x +8 dx x (x + ) + dx tan tan + sec d tan sec sec d tan sec sec d (tan )(sec ) d (sec tan sec ) d (sec ln sec + tan )+C Now use that x + tan to construct the following triangle. (x + ) + x + (x + ) The triangle gives sec +. Therefore, x (x + ) x +x +8 dx + ln (x + ) + + x +! + C Work on Problem: Comletes the square Defines x + tan Finds dx Rewrites the integral using the trig sub Uses the trig identity + tan sec Simlifies the integral Finds the antiderivatives of sec tan and sec Uses the triangle to write the final answer in terms of x with + C Notes: Deduct.5 oints for each notation error (with a maximum deduction of oints for notation errors) Points oints oints oints oint oints oints oints ( oint each) oints - Page 8 of 5

9 MATH 8 3. ( ts.) Evaluate the integral: Solution (Method ): Test - Answer Key tan 3 x sec x dx. Sring 6 Sections , tan 3 x / sec x dx sin 3 x cos 3 x cos x dx Now use u-substitution. Let u cos x. Thendu sin 3 x cos 3 x cos xdx sin 3 x cos x dx sin x sin x cos x dx ( cos x)sinx cos x sin xdx.so dx ( cos x)sinx cos x dx / / ln u ln u ( u ) du u u du u / ln So tan 3 x sec x dx ln - Page 9 of 5

10 MATH 8 Test - Answer Key Sring 6 Sections , Work on Problem: Rewrites in terms of sines and cosines Simlifies Uses an aroriate Pythagorean identity No credit is given for a wrong identity Defines u and finds du Rewrites the integral in terms of u with u-limits oints for writing in terms of u/du, oint for limits Finds the antiderivative of each term Evaluates Notes: Deduct.5 oints for notation errors Deduct.5 oints for leaving ln in answer Deduct.5 oints for mixing x and u in integral Deduct.5 oints for additional incorrect trig identities Points oints oint 3 oints oints ( oint each) 3 oints oints oint - Page of 5

11 MATH 8 Test - Answer Key Sring 6 Sections , Solution (Method ): tan 3 x / sec x dx tan x tan x sec x dx (sec ) tan x sec dx x tan x tan xdx tan xdx tan x sec x dx sin x cos x cos x dx sin x cos xdx Now use u-substitution. Let u sinx. Thendu cos xdx.so So tan xdx tan 3 x sec x dx ln( ) sin x cos xdx tan xdx ln sec x / dx ln( ) / u udu / - Page of 5

12 MATH 8 Test - Answer Key Sring 6 Sections , Solution (Method 3): tan 3 x sec x dx tan 3 x sec xdx tan x sec x(tan x)sec 3 xdx tan x sec x(sec x ) sec 3 xdx Now use u-substitution. Let u secx. Thendu secx tan xdx.so tan x sec x(sec x ) sec 3 xdx ln u (u )u 3 du u u ln u + u ln( ) u 3 du So tan 3 x sec x dx ln( ) - Page of 5

13 MATH 8 Test - Answer Key Sring 6 Sections , ( ts.) A 5-ft chain with a -ound load attached to it hangs from a rod that is 5 ft above the ground. Set u, but do not evaluate or simlify, the integral that reresents the work done in winding u the entire chain (with the load attached) onto the rod if the chain weighs 3 ounds er foot. Solution: Let x the distance from the bottom of the chain to the ground, W T total work done, W C work to lift the chain, W L work to lift the load. Then W L ( lbs)(5 ft) 5 ft-lbs and W C 5 3(5 x) dx So W T (5 x) dx ft-lbs Work on Problem: Finds the work to lift the load Sets u an integral to find the work to lift the chain Sums u both arts to exress the total work done Notes: Deduct.5 oints notation errors. Do not deduct for missing units. Points 3 oints 6 oints oint - Page 3 of 5

14 MATH 8 Test - Answer Key Sring 6 Sections , (8 ts.) Let R be the region bounded by y e x, y, and the y-axis. (a) Set u, but do not evaluate or simlify, the integral that gives the volume of the solid obtained by rotating the region R around the y-axis using the disk/washer method. Solution: V ln y dy (b) Set u, but do not evaluate or simlify, the integral that gives the volume of the solid obtained by rotating the region R around the line y using the disk/washer method. Solution: V ln h 5 e x + i dx (c) Set u, but do not evaluate or simlify, the integral that gives the volume of the solid obtained by rotating the region R around the line x using the shell method. Solution: V ln ( x)( e x ) dx Work on each Problem: Limits of integration Aroriate constant ( on (a), (b); on (c)) Rest of integrand Notes: Deduct.5 oints for each notation error (with maximum deduction of oint for notation errors) Points oint oint oints - Page of 5

15 MATH 8 Test - Answer Key Sring 6 Sections , 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 8 bubbled in answers; has MATH 8 and my Section number written at the to; has my Instructor s last name written at the to; has Test No. written at the to; has the correct test version written at the to 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 5 of 5