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Unformatted text preview: Vaughn, Laura – Final 1 – Due: May 11 2007, 11:00 pm – Inst: Katja Mallmann 1 This printout should have 24 questions. Multiplechoice 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 thRiemann sum approximation to the definite integral I = Z b a f ( x ) dx is given by n X i = 1 f ( x * i ) Δ 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: Vaughn, Laura – Final 1 – Due: May 11 2007, 11:00 pm – Inst: Katja Mallmann 2 By definition, Z b a f ( x ) dx = lim n → ∞ f ( x * i ) Δ 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: Vaughn, Laura – Final 1 – Due: May 11 2007, 11:00 pm – Inst: Katja Mallmann 3 Since the car leaves Austin at time t = 0, the position of the car t hours later is the antiderivative 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.
 Spring '08
 RAdin

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