M408L hwk5 solutions - Pham, Quoc Homework 5 Due: Feb 20...

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Unformatted text preview: Pham, Quoc Homework 5 Due: Feb 20 2007, 3:00 am Inst: Eric Katerman 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Find the most general function f such that f ( x ) = 6 + 5 4- x 2 . 1. f ( x ) = 3- 5 2 tan- 1 x 2 + C 2. f ( x ) = 6 x + 5 2 tan- 1 x 2 + C 3. f ( x ) = 6- 5 2 tan- 1 x + C 4. f ( x ) = 6 x + 5 sin- 1 x 2 + C correct 5. f ( x ) = 6 x- 5 sin- 1 x 2 + C 6. f ( x ) = 3 x + 5 sin- 1 x + C Explanation: Since d dx sin- 1 x 2 = 1 4- x 2 , we see that f ( x ) = 6 x + 5 sin- 1 x 2 + C with C an arbitrary constant. keywords: 002 (part 1 of 1) 10 points Find the value of the integral I = Z 4 1 4 9 + ( x- 1) 2 dx. 1. I = 2 3 2. I = 3 3. I = 2 3 4. I = 3 5. I = 1 3 correct 6. I = 4 3 Explanation: Set 3 tan u = x- 1. Then 9 + ( x- 1) 2 = 9 + (3 tan u ) 2 = 9(1 + tan 2 u ) = 9 sec 2 u, while 3 sec 2 udu = dx. Also x = 1 = u = 0 , and x = 4 = u = 4 . In this case I = Z / 4 12 sec 2 u 9 sec 2 u du = 4 3 Z / 4 du. Consequently, I = 4 3 h u i / 4 = 1 3 . keywords: 003 (part 1 of 1) 10 points Evaluate the definite integral I = Z 1 4 1 1- 4 x 2 dx. Correct answer: 0 . 261799 . Explanation: Pham, Quoc Homework 5 Due: Feb 20 2007, 3:00 am Inst: Eric Katerman 2 Since Z 1 1- x 2 dx = sin- 1 x + C , a change of variable x is needed to reduce I to this form. Set u = 2 x . Then du = 2 dx , and x = 0 = u = 0 , while x = 1 4 = u = 1 2 . In this case I = 1 2 Z 1 2 1 1- u 2 du = 1 2 sin- 1 u 1 2 . Consequently, I = 1 2 arcsin 1 2 = 0 . 261799 . keywords: 004 (part 1 of 1) 10 points Find the value of the definite integral I = Z 3 2 8 sin- 1 x 1- x 2 dx. 1. I = 4 25 2 2. I = 1 3 2 3. I = 1 9 2 4. I = 4 9 2 correct 5. I = 1 4 2 Explanation: Since Z 1 1- x 2 dx = sin- 1 x + C , this suggests the substitution u = sin- 1 x , for then du = 1 1- x 2 dx, while x = 0 = u = 0 , x = 3 2 = u = 3 ....
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This homework help was uploaded on 03/19/2008 for the course M 408L taught by Professor Radin during the Spring '08 term at University of Texas at Austin.

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M408L hwk5 solutions - Pham, Quoc Homework 5 Due: Feb 20...

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