Final1 Solutions Sp '07

# Final1 Solutions Sp '07 - Lemmons, Amanda Final 1 Due: May...

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Unformatted text preview: Lemmons, Amanda Final 1 Due: May 12 2007, 1:00 am Inst: Mar Gonzalez 1 This print-out should have 24 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 When f has graph R 1 R 2 a b c express the sum I = Z c a n 4 f ( x )-| f ( x ) | o dx in terms of the areas A 1 = area( R 1 ) , A 2 = area( R 2 ) of the respective lighter shaded regions R 1 and R 2 . 1. I = 3 A 1- 5 A 2 correct 2. I =- 5 A 2 3. I = 3 A 1 4. I =- 3 A 1- 5 A 2 5. I = 3 A 1 + 5 A 2 6. I =- 3 A 1 + 5 A 2 Explanation: As the graph shows, f takes positive values on [ a, b ] and negative values on [ b, c ], so Z b a f ( x ) dx = A 1 , Z c b f ( x ) dx =- A 2 . while Z c a | f ( x ) | dx = Z b a f ( x ) dx- Z c b f ( x ) dx, and Z c a f ( x ) dx = Z b a f ( x ) dx + Z c b f ( x ) dx, Thus I = 4( A 1- A 2 )- ( A 1 + A 2 ) . Consequently, I = 3 A 1- 5 A 2 . keywords: definite integral, properties inte- grals, area 002 (part 1 of 1) 10 points If an n th-Riemann sum approximation to the definite integral I = Z b a f ( x ) dx is given by n X i = 1 f ( x * i ) x i = n 2- 6 n + 4 6 n 2 , determine the value of I . 1. I = 1 6 correct 2. I =- 1 6 3. I = 2 3 4. I =- 5 6 5. I =- 1 Explanation: Lemmons, Amanda Final 1 Due: May 12 2007, 1:00 am Inst: Mar Gonzalez 2 By definition, Z b a f ( x ) dx = lim n f ( x * i ) x i . Thus Z b a f ( x ) dx = lim n n 2- 6 n + 4 6 n 2 . Consequently, I = 1 6 . keywords: Riemann sum, definition definite integral, limit, conceptual 003 (part 1 of 1) 10 points Determine F ( x ) when F ( x ) = Z x 5 8 sin t t dt. 1. F ( x ) =- 8 cos x x 2. F ( x ) =- 8 cos x x 3. F ( x ) = 4 sin( x ) x correct 4. F ( x ) =- 8 cos( x ) x 5. F ( x ) =- 4 sin( x ) x 6. F ( x ) = 8 sin x x 7. F ( x ) = 4 sin x x 8. F ( x ) = 4 cos( x ) x Explanation: By the Fundamental Theorem of Calculus and the Chain Rule, d dx Z g ( x ) a f ( t ) dt = f ( g ( x )) g ( x ) . When F ( x ) = Z x 5 8 sin t t dt, therefore, F ( x ) = 8 sin( x ) x d dx x . Consequently, F ( x ) = 4 sin( x ) x , since d dx x = 1 2 x . keywords: Stewart5e, FTC, Chain Rule 004 (part 1 of 1) 10 points A car heads north from Austin on IH 35. Its velocity t hours after leaving Austin is given in miles per hour by v ( t ) = 42 + 22 3 t- t 2 . Determine how many hours will elapse before the car is next in Austin. 1. 19 hours later 2. 17 hours later 3. car never returns to Austin 4. 20 hours later 5. 18 hours later correct Explanation: Lemmons, Amanda Final 1 Due: May 12 2007, 1:00 am Inst: Mar Gonzalez 3 Since the car leaves Austin at time t = 0, the position of the car t hours later is the anti-derivative s ( t ) = Z (42 + 22 3 t- t 2 ) dt, s (0) = 0 of v ( t ). But Z (42 + 22 3 t- t 2 ) dt = 42 t + 11 3 t 2- 1 3 t 3 = t ( t + 7) 6- 1 3 t + C....
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## This note was uploaded on 03/20/2008 for the course M 408L taught by Professor Radin during the Spring '08 term at University of Texas at Austin.

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Final1 Solutions Sp '07 - Lemmons, Amanda Final 1 Due: May...

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