HW_9_Key - Miller, Kierste Homework 9 Due: Oct 26 2007,...

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Unformatted text preview: Miller, Kierste Homework 9 Due: Oct 26 2007, 3:00 am Inst: JEGilbert 1 This print-out should have 16 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 Determine the vector I = Z 1 r ( t ) dt when r ( t ) = D 4 1 + t 2 , 2 t 1 + t 2 , 2 (1 + t ) 2 E . 1. I = h 4 , ln2 , 2 i 2. I = h ln2 , , 1 i 3. I = h , ln2 , 1 i correct 4. I = h , 2 , ln2 i 5. I = h 4 , 2 , 2ln2 i 6. I = h 4ln2 , 2 , 2 i Explanation: For a vector function r ( t ) = h f ( t ) , g ( t ) , h ( t ) i , the components of the vector I = Z 1 r ( t ) dt are given by Z 1 f ( t ) dt, Z 1 g ( t ) dt, Z 1 h ( t ) dt, respectively. But when r ( t ) = D 4 1 + t 2 , 2 t 1 + t 2 , 2 (1 + t ) 2 E , we see that Z 1 f ( t ) dt = Z 1 4 1 + t 2 dt = h 4tan- 1 t i 1 = , while Z 1 g ( t ) dt = Z 1 2 t 1 + t 2 dt = h ln(1 + t 2 ) i 1 = ln2 , and Z 1 h ( t ) dt = Z 1 2 (1 + t ) 2 dt = h- 2 1 + t i 1 = 1 . Consequently, I = h , ln2 , 1 i . keywords: vector function, definite integral, inverse trig integral, log function, substitution 002 (part 1 of 1) 10 points Find the arc length of the curve r ( t ) = h sin2 t, 6 t, cos2 t i between r (0) and r (3). 1. arc length = 3 37 2. arc length = 6 10 correct 3. arc length = 18 4. arc length = 3 38 5. arc length = 3 39 Explanation: The length of the curve r ( t ) between r ( t ) and r ( t 1 ) is given by the integral L = Z t 1 t | r ( t ) | dt. Miller, Kierste Homework 9 Due: Oct 26 2007, 3:00 am Inst: JEGilbert 2 Now when r ( t ) = h sin2 t, 6 t, cos2 t i , we see that r ( t ) = h 2cos2 t, 6 ,- 2sin2 t i . But then by the Pythagorean identity, | r ( t ) | = (4 + 36) 1 / 2 . Thus L = Z 3 2 10 dt = h 2 10 t i 3 . Consequently, arc length = L = 6 10 . keywords: curve, 3-space, length, trig func- tion 003 (part 1 of 1) 10 points Find the arc length of the curve r ( t ) = (4 + 2 t ) i + e t j + (1 + e- t ) k between r (0) and r (4). 1. arc length = 2 e 4 2. arc length = e 4 + e- 4 3. arc length = e 4- e- 4 correct 4. arc length = ( e 4- e- 4 ) 2 5. arc length = ( e 4 + e- 4 ) 2 6. arc length = 2 e- 4 Explanation: The length of a curve r ( t ) between r ( t ) and r ( t 1 ) is given by the integral L = Z t 1 t | r ( t ) | dt. Now when r ( t ) = (4 + 2 t ) i + e t j + (1 + e- t ) k , we see that r ( t ) = 2 i + e t- e- t . But then | r ( t ) | = (2 + e 2 t + e- 2 t ) 1 / 2 = e t + e- t . Thus L = Z 4 ( e t + e- t ) dt = h e t- e- t i 4 . Consequently, arc length = L = e 4- e- 4 . keywords: curve, 3-space, arc length, exp function 004 (part 1 of 1) 10 points Find the unit vector T ( t ) tangent to the curve r ( t ) = 6 t 2 , 12 t, 6ln t fi ....
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This note was uploaded on 05/08/2008 for the course M 408 M taught by Professor Gilbert during the Fall '07 term at University of Texas at Austin.

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HW_9_Key - Miller, Kierste Homework 9 Due: Oct 26 2007,...

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