# HW 3 - Portillo, Tom Homework 3 Due: Sep 19 2006, 3:00 am...

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Portillo, Tom – 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 = - 7 3. x = - 1 4. not enough inFormation given 5. x = 2 . 5 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 (2 + cos t ) dt . 1. g 0 ( x ) = 2 - sin x 2. g 0 ( x ) = 2 x + sin x 3. g 0 ( x ) = 2 + cos x correct 4. g 0 ( x ) = - sin x 5. g 0 ( x ) = 2 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 ) = 2 + cos x . keywords:

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Portillo, Tom – Homework 3 – Due: Sep 19 2006, 3:00 am – Inst: David Benzvi 2 003 (part 1 of 1) 10 points If F ( x ) = Z x 0 e 6 sin 2 θ dθ , Fnd the value of F 0 ( π/ 4). 1. F 0 ( π/ 4) = e 3 correct 2. F 0 ( π/ 4) = 3 e 3. F 0 ( π/ 4) = 3 e 3 4. F 0 ( π/ 4) = 3 e 6 5. F 0 ( π/ 4) = e 6 Explanation: By the ±undamental theorem of calculus, F 0 ( x ) = e 6 sin 2 x . At x = π/ 4, therefore, F 0 ( π/ 4) = 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 + 3 sin x ) dx . 1. I = 8 2. I = 6 3. I = 9 4. I = 7 correct 5. I = 10 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 + 3 sin x . Taking F ( x ) = 4 sin x - 3 cos x and using the fact that cos 0 = sin π 2 = 1 , sin 0 = cos π 2 = 0 , we thus see that I = 7 . keywords: integral, ±TC, trig function 005 (part 1 of 1) 10 points Evaluate the deFnite integral I = Z 1 0 (1 + 4 x - 2 x 2 ) dx . Correct answer: 2 . 33333 . Explanation: By linearity, Z 1 0 (1 + 4 x - 2 x 2 ) dx = Z 1 0 dx + 4 Z 1 0 x dx - 2 Z 1 0 x 2 dx . But Z 1 0 x r dx = h x r +1 r + 1 i 1 0 = 1 r + 1 ,
Portillo, Tom – Homework 3 – Due: Sep 19 2006, 3:00 am – Inst: David Benzvi 3 provided r 6 = - 1. Consequently, I = 7 3 = 2 . 33333 .

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## This note was uploaded on 02/18/2010 for the course MATH 408L taught by Professor Benzvi during the Spring '06 term at North Texas.

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HW 3 - Portillo, Tom Homework 3 Due: Sep 19 2006, 3:00 am...

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