Exam1Sol

Exam1Sol - Massaro Michael Exam 1 Due Oct 2 2007 11:00 pm...

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Massaro, Michael – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: Shinko Harper 1 This print-out should have 18 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 Below is the graph of a function f . 2 4 6 8 10 12 2 4 6 8 - 2 - 4 - 6 Estimate the definite integral I = Z 12 0 f ( x ) dx with six equal subintervals using left end- points. 1. I 10 2. I 6 3. I 8 4. I 14 correct 5. I 12 Explanation: Since [0 , 12] is subdivided into six equal subintervals, each of these will have length 2 and the six corresponding rectangles are shown as the shaded areas in 2 4 6 8 10 12 2 4 6 8 - 2 - 4 - 6 The heights of the rectangles are left endpoint sample values of f that can be read off from the graph. Thus, with left endpoints, I 2(4 + 2 - 4 - 1 + 2 + 4) = 14 . keywords: graph, Riemann sum, left end- points 002 (part 1 of 1) 10 points If F ( x ) = Z x 0 2 e 10 sin 2 θ dθ , find the value of F 0 ( π/ 4). 1. F 0 ( π/ 4) = 5 e 2 2. F 0 ( π/ 4) = 5 e 5 3. F 0 ( π/ 4) = 2 e 10 4. F 0 ( π/ 4) = 2 e 5 correct 5. F 0 ( π/ 4) = 5 e 10 Explanation: By the Fundamental theorem of calculus, F 0 ( x ) = 2 e 10 sin 2 x . At x = π/ 4, therefore, F 0 ( π/ 4) = 2 e 5

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Massaro, Michael – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: Shinko Harper 2 since sin( π 4 ) = 1 2 . keywords: integral, FTC 003 (part 1 of 1) 10 points Determine F 0 ( x ) when F ( x ) = Z x 4 4 sin t t dt . 1. F 0 ( x ) = 2 sin( x ) x correct 2. F 0 ( x ) = 2 sin x x 3. F 0 ( x ) = 4 sin x x 4. F 0 ( x ) = 2 cos( x ) x 5. F 0 ( x ) = - 4 cos x x 6. F 0 ( x ) = - 4 cos x x 7. F 0 ( x ) = - 2 sin( x ) x 8. F 0 ( 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 0 ( x ) . When F ( x ) = Z x 4 4 sin t t dt , therefore, F 0 ( x ) = 4 sin( x ) x d dx x · . Consequently, F 0 ( x ) = 2 sin( x ) x , since d dx x = 1 2 x . keywords: Stewart5e, FTC, Chain Rule 004 (part 1 of 1) 10 points If w 0 ( t ) is the rate of growth of Mira’s weight (in pounds per year), what does the definite integral I = Z 8 2 w 0 ( t ) dt represent? 1. Mira’s weight at age 8 2. average of Mira’s weight from age 2 to 8 3. increase in Mira’s weight from age 2 to 8 correct 4. Mira’s weight at age 2 5. decrease in Mira’s weight from age 2 to 8 Explanation: By the Fundamental theorem of Calculus, Z b a w 0 ( x ) dx = w ( b ) - w ( a ) , in other words, the value of the integral is the net change, w ( b ) - w ( a ), in w over the interval [ a, b ]. Consequently, I is the increase in Mira’s weight from age 2 to 8 .
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