AnsHW3 - Cheung Anthony Homework 3 Due 3:00 am Inst David...

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Cheung, Anthony – Homework 3 – Due: Sep 19 2006, 3:00 am – Inst: David Benzvi 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. The due time is Central time. 001 (part 1 oF 1) 10 points The graph oF f is shown in the fgure 2 4 6 8 2 4 - 2 IF the Function g is defned by g ( x ) = Z x 1 f ( t ) dt, For what value oF x does g ( x ) have a maxi- mum? 1. x = 5 correct 2. x = - 1 3. x = 2 . 5 4. x = - 7 5. not enough inFormation given Explanation: By the ±undamental theorem oF calculus, iF g ( x ) = Z x 1 f ( t ) dt, then g 0 ( x ) = f ( x ). Thus the critical points oF g occur at the zeros oF f , i.e. , at the x - intercepts oF the graph oF f . To determine which oF these gives a local maximum oF g we use the sign chart g 0 + - 1 5 7 For g 0 . This shows that max g ( x ) at x = 5 , since the sign oF g 0 changes From positive to negative at x = 5. keywords: ±TC, integral, sign chart, maxi- mum 002 (part 1 oF 1) 10 points ±ind g 0 ( x ) when g ( x ) = Z x π (8 + cos t ) dt . 1. g 0 ( x ) = 8 x + sin x 2. g 0 ( x ) = 8 - sin x 3. g 0 ( x ) = - sin x 4. g 0 ( x ) = 8 + cos x correct 5. g 0 ( x ) = 8 x - cos x Explanation: By the ±undamental theorem oF Calculus, iF g ( x ) = Z x a f ( t ) dt , then g 0 ( x ) = d dx Z x a g ( t ) dt = f ( x ) . In the given example, thereFore, g 0 ( x ) = 8 + cos x . keywords:
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2 003 (part 1 of 1) 10 points If F ( x ) = Z x 0 2 e 6 sin 2 θ dθ , Fnd the value of F 0 ( π/ 4). 1. F 0 ( π/ 4) = 3 e 2 2. F 0 ( π/ 4) = 3 e 6 3. F 0 ( π/ 4) = 2 e 6 4. F 0 ( π/ 4) = 2 e 3 correct 5. F 0 ( π/ 4) = 3 e 3 Explanation: By the ±undamental theorem of calculus, F 0 ( x ) = 2 e 6 sin 2 x . At x = π/ 4, therefore, F 0 ( π/ 4) = 2 e 3 since sin( π 4 ) = 1 2 . keywords: integral, ±TC 004 (part 1 of 1) 10 points Evaluate the deFnite integral I = Z π/ 2 0 (4 cos x - sin x ) dx . 1. I = 1 2. I = 4 3. I = 2 4. I = 5 5. I = 3 correct Explanation: By the ±undamental Theorem of Calculus, I = h F ( x ) i π/ 2 0 = F ( π 2 ) - F (0) for any anti-derivative F of f ( x ) = 4 cos x - sin x . Taking F ( x ) = 4 sin x + cos x and using the fact that cos 0 = sin π 2 = 1 , sin 0 = cos π 2 = 0 , we thus see that I = 3 . keywords: integral, ±TC, trig function 005 (part 1 of 1) 10 points Evaluate the deFnite integral I = Z 1 0 (5 + 8 x - 9 x 2 ) dx . Correct answer: 6 . Explanation: By linearity, Z 1 0 (5 + 8 x - 9 x 2 ) dx = 5 Z 1 0 dx + 8 Z 1 0 x dx - 9 Z 1 0 x 2 dx . But
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AnsHW3 - Cheung Anthony Homework 3 Due 3:00 am Inst David...

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