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Unformatted text preview: Kang, Danby – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: Diane Radin 1 This printout should have 18 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 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 f ( x ) dx with six equal subintervals using left end points. 1. I ≈ 10 2. I ≈ 14 3. I ≈ 12 4. I ≈ 16 5. I ≈ 18 correct 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(5 + 2 4 1 + 3 + 4) = 18 . keywords: graph, Riemann sum, left end points 002 (part 1 of 1) 10 points If F ( x ) = Z x 6 e 4 sin 2 θ dθ , find the value of F ( π/ 4). 1. F ( π/ 4) = 6 e 2 correct 2. F ( π/ 4) = 2 e 6 3. F ( π/ 4) = 6 e 4 4. F ( π/ 4) = 2 e 2 5. F ( π/ 4) = 2 e 4 Explanation: By the Fundamental theorem of calculus, F ( x ) = 6 e 4 sin 2 x . At x = π/ 4, therefore, F ( π/ 4) = 6 e 2 Kang, Danby – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: Diane Radin 2 since sin( π 4 ) = 1 √ 2 . keywords: integral, FTC 003 (part 1 of 1) 10 points Determine F ( x ) when F ( x ) = Z √ x 4 4 cos t t dt. 1. F ( x ) = 2 cos x x 2. F ( x ) = 2 sin( √ x ) x 3. F ( x ) = 4 sin x x 4. F ( x ) = 2 cos( √ x ) √ x 5. F ( x ) = 2 cos( √ x ) x correct 6. F ( x ) = 4 cos x √ x 7. F ( x ) = 4 sin x √ x 8. F ( x ) = 4 sin( √ 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 4 4 cos t t dt, therefore, F ( x ) = 4 cos( √ x ) √ x ‡ d dx √ x · . Consequently, F ( x ) = 2 cos( √ x ) x , since d dx √ x = 1 2 √ x . keywords: Stewart5e, FTC, Chain Rule 004 (part 1 of 1) 10 points If w ( t ) is the rate of growth of Mira’s weight (in pounds per year), what does the definite integral I = Z 5 2 w ( t ) dt represent? 1. increase in Mira’s weight from age 2 to 5 correct 2. average of Mira’s weight from age 2 to 5 3. Mira’s weight at age 2 4. Mira’s weight at age 5 5. decrease in Mira’s weight from age 2 to 5 Explanation: By the Fundamental theorem of Calculus, Z b a w ( 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 ]....
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 Fall '08
 Cepparo
 Calculus, dx, Diane Radin

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