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solution final_pdf - Version 004 FINAL Odell(58340 This...

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Version 004 – FINAL – Odell – (58340) 1 This print-out should have 24 questions. Multiple-choice 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 ) = cos 1 3 t + 4 sin 2 3 t and f (0) = - 1. 1. f ( π/ 2) = 11 2 2. f ( π/ 2) = 13 2 3. f ( π/ 2) = 5 2 4. f ( π/ 2) = 9 2 5. f ( π/ 2) = 7 2 correct Explanation: The function f must have the form f ( t ) = 3 sin 1 3 t - 6 cos 2 3 t + C where the constant C is determined by the condition f (0) = 3 sin 0 - 6 cos 0 + C = - 1 . Thus f ( t ) = 3 sin 1 3 t - 6 cos 2 3 t + 5 . Consequently, f ( π/ 2) = 7 2 . 002 10.0 points When f has graph R 2 R 1 c b a express the sum I = integraldisplay c a braceleftBig f ( x ) - 4 | 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 = - 5 A 1 + 3 A 2 2. I = - 3 A 1 3. I = - 5 A 1 - 3 A 2 correct 4. I = 5 A 1 - 3 A 2 5. I = 5 A 1 + 3 A 2 6. I = - 5 A 2 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 ,
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Version 004 – FINAL – Odell – (58340) 2 Thus I = ( - A 1 + A 2 ) - 4( A 1 + A 2 ) . Consequently, I = - 5 A 1 - 3 A 2 . 003 10.0 points Evaluate the definite integral I = integraldisplay 1 0 parenleftBig 8 x - x 1 / 3 parenrightBig dx . 1. I = 15 4 2. I = 13 4 correct 3. I = 7 2 4. I = 7 3 5. I = 2 6. I = 8 3 Explanation: By the Fundamental Theorem of Calculus, I = bracketleftBig F ( x ) bracketrightBig 1 0 = F (1) - F (0) for any anti-derivative F of f ( x ) = 8 x - x 1 / 3 . Taking F ( x ) = 4 x 2 - 3 4 x 4 / 3 , we thus see that I = 4 - 3 4 . Consequently, I = 13 4 . 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 ) = 4 - 3 t + 3 10 t 2 What will be the position of the car after 10 hours of driving? 1. 70 miles south of Austin 2. 70 miles north of Austin 3. 10 miles south of Austin correct 4. 90 miles north of Austin 5. 90 miles south of Austin 6. 30 miles south of Austin 7. 10 miles north of Austin 8. 30 miles north of Austin Explanation: Since the car leaves Austin at time t = 0, its position t hours later is the anti-derivative s ( t ) = integraldisplay (4 - 3 t + 3 10 t 2 ) dt, s (0) = 0 of v ( t ). But integraldisplay (4 - 3 t + 3 10 t 2 ) dt = 4 t - 3 2 t 2 + 1 10 t 3 + C . On the other hand, s (0) = 0 = C = 0 . Thus s ( t ) = 4 t - 3 2 t 2 + 1 10 t 3 .
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Version 004 – FINAL – Odell – (58340) 3 At t = 10, therefore, s (10) = - 10 , the negative sign indicating that the car is 10 miles south of Austin after 10 hours of driving. keywords: velocity, position, definite integral, quadratic function 005 10.0 points Evaluate the integral I = integraldisplay 3 0 d dx (4 + 3 x 2 ) 1 / 2 dx. 1. I = 31 - 2 correct 2. I = 2 - 31 3. I = 31 4. I = 2 5. I = 31 + 2 Explanation: As an indefinite integral, integraldisplay d dx (4 + 3 x 2 ) 1 / 2 dx = (4 + 3 x 2 ) 1 / 2 + C where C is an arbitrary constant. Thus integraldisplay 3 0 d dx (4 + 3 x 2 ) 1 / 2 dx = bracketleftBig (4 + 3 x 2 ) 1 / 2 bracketrightBig 3 0 .
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