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Unformatted text preview: Version 100 – FINAL – Gompf – (58370) 1 This printout should have 24 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find f ( π/ 2) when f ′ ( t ) = 3 cos 1 3 t 4 sin 2 3 t and f (0) = 1. 1. f ( π/ 2) = 1 2 2. f ( π/ 2) = 1 2 3. f ( π/ 2) = 5 2 correct 4. f ( π/ 2) = 3 2 5. f ( π/ 2) = 3 2 Explanation: The function f must have the form f ( t ) = 9 sin 1 3 t + 6 cos 2 3 t + C where the constant C is determined by the condition f (0) = 9 sin0 + 6 cos0 + C = 1 . Thus f ( t ) = 9 sin 1 3 t + 6 cos 2 3 t 5 . Consequently, f ( π/ 2) = 5 2 . 002 10.0 points When f has graph R 2 R 1 c b a express the sum I = integraldisplay c a braceleftBig 2 f ( x )  f ( x )  bracerightBig 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 + A 2 2. I = 3 A 2 3. I = 3 A 1 A 2 4. I = 3 A 1 A 2 5. I = A 1 6. I = 3 A 1 + A 2 correct Explanation: As the graph shows, f takes negative values on [ a, b ] and positive values on [ b, c ], so integraldisplay b a f ( x ) dx = A 1 , integraldisplay c b f ( x ) dx = A 2 . while integraldisplay c a  f ( x )  dx = integraldisplay b a f ( x ) dx + integraldisplay c b f ( x ) dx , and integraldisplay c a f ( x ) dx = integraldisplay b a f ( x ) dx + integraldisplay c b f ( x ) dx , Version 100 – FINAL – Gompf – (58370) 2 Thus I = 2( A 1 + A 2 ) ( A 1 + A 2 ) . Consequently, I = 3 A 1 + A 2 . 003 10.0 points Evaluate the definite integral I = integraldisplay 1 parenleftBig 8 x x 2 / 3 parenrightBig dx . 1. I = 18 5 2. I = 2 3. I = 7 3 4. I = 19 5 5. I = 5 3 6. I = 17 5 correct Explanation: By the Fundamental Theorem of Calculus, I = bracketleftBig F ( x ) bracketrightBig 1 = F (1) F (0) for any antiderivative F of f ( x ) = 8 x x 2 / 3 . Taking F ( x ) = 4 x 2 3 5 x 5 / 3 , we thus see that I = 4 3 5 . Consequently, I = 17 5 . 004 10.0 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 ) = 3 2 t 3 10 t 2 What will be the position of the car after 10 hours of driving? 1. 230 miles south of Austin 2. 185 miles north of Austin 3. 230 miles north of Austin 4. 185 miles south of Austin 5. 170 miles south of Austin 6. 215 miles north of Austin 7. 215 miles south of Austin 8. 170 miles north of Austin correct Explanation: Since the car leaves Austin at time t = 0, its position t hours later is the antiderivative s ( t ) = integraldisplay (3 2 t 3 10 t 2 ) dt, s (0) = 0 of v ( t ). But integraldisplay (3 2 t 3 10 t 2 ) dt = 3 t t 2 1 10 t 3 + C . On the other hand, s (0) = 0 = ⇒ C = 0 ....
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This note was uploaded on 12/04/2009 for the course M 408L taught by Professor Radin during the Spring '08 term at University of Texas at Austin.
 Spring '08
 RAdin
 Calculus

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