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Unformatted text preview: Valenica, Daniel – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: Cheng 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 ≈ 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 5 e 8 sin 2 θ dθ , find the value of F ( π/ 4). 1. F ( π/ 4) = 4 e 4 2. F ( π/ 4) = 4 e 5 3. F ( π/ 4) = 4 e 8 4. F ( π/ 4) = 5 e 4 correct 5. F ( π/ 4) = 5 e 8 Explanation: By the Fundamental theorem of calculus, F ( x ) = 5 e 8 sin 2 x . At x = π/ 4, therefore, F ( π/ 4) = 5 e 4 Valenica, Daniel – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: Cheng 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 5 8 sin t t dt. 1. F ( x ) = 8 cos x x 2. F ( x ) = 8 cos x √ x 3. F ( x ) = 4 sin( √ x ) x correct 4. F ( x ) = 8 cos( √ x ) √ x 5. F ( x ) = 4 sin( √ x ) √ x 6. F ( x ) = 8 sin x √ x 7. F ( x ) = 4 sin x x 8. F ( 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 ( x ) . When F ( x ) = Z √ x 5 8 sin t t dt, therefore, F ( x ) = 8 sin( √ x ) √ x ‡ d dx √ x · . Consequently, F ( 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 ( t ) is the rate of growth of Mira’s weight (in pounds per year), what does the definite integral I = Z 8 3 w ( 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 ( 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 '07
 CHENG
 Calculus, Sin, dx, Valenica

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