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Integral Calculus Exam 1

# Integral Calculus Exam 1 - Panjwani Sameer Exam 1 Due Oct 2...

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Panjwani, Sameer – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: James Rath 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 14 correct 2. I 6 3. I 10 4. I 12 5. I 8 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 + 1 - 4 - 1 + 3 + 4) = 14 . keywords: graph, Riemann sum, left end- points 002 (part 1 of 1) 10 points If F ( x ) = Z x 0 5 e 8 sin 2 θ dθ , find the value of F 0 ( π/ 4). 1. F 0 ( π/ 4) = 4 e 4 2. F 0 ( π/ 4) = 4 e 5 3. F 0 ( π/ 4) = 4 e 8 4. F 0 ( π/ 4) = 5 e 4 correct 5. F 0 ( π/ 4) = 5 e 8 Explanation: By the Fundamental theorem of calculus, F 0 ( x ) = 5 e 8 sin 2 x . At x = π/ 4, therefore, F 0 ( π/ 4) = 5 e 4

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Panjwani, Sameer – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: James Rath 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 5 8 sin t t dt . 1. F 0 ( x ) = - 8 cos x x 2. F 0 ( x ) = - 8 cos x x 3. F 0 ( x ) = 4 sin( x ) x correct 4. F 0 ( x ) = - 8 cos( x ) x 5. F 0 ( x ) = - 4 sin( x ) x 6. F 0 ( x ) = 8 sin x x 7. F 0 ( x ) = 4 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 5 8 sin t t dt , therefore, F 0 ( x ) = 8 sin( x ) x d dx x · . Consequently, F 0 ( x ) = 4 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 3 w 0 ( t ) dt represent? 1. decrease in Mira’s weight from age 3 to 8 2. average of Mira’s weight from age 3 to 8 3. Mira’s weight at age 8 4. increase in Mira’s weight from age 3 to 8 correct 5. Mira’s weight at age 3 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 3 to 8 . keywords: integral, rate growth, FTC, weight, net change, 005 (part 1 of 1) 10 points
Panjwani, Sameer – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: James Rath 3 Find the value of f (0) when f 00 ( t ) = 6(2 t + 1) and f 0 (1) = 6 , f (1) = 2 .

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