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homework 3 solutions

# homework 3 solutions - Mcbiles Emily Homework 3 Due 3:00 am...

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Mcbiles, Emily – Homework 3 – Due: Sep 19 2006, 3:00 am – Inst: David Fonken 1 This print-out should have 22 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 The graph of f is shown in the figure -2 -1 0 1 2 3 4 5 6 7 8 9 2 4 6 8 2 4 - 2 If the function g is defined by g ( x ) = Z x 2 f ( t ) dt, for what value of x does g ( x ) have a maxi- mum? 1. x = - 8 2. not enough information given 3. x = 3 . 5 4. x = - 2 5. x = 6 correct Explanation: By the Fundamental theorem of calculus, if g ( x ) = Z x 2 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 + - 2 6 8 for g 0 . This shows that max g ( x ) at x = 6 , since the sign of g 0 changes from positive to negative at x = 6. keywords: FTC, integral, sign chart, maxi- mum 002 (part 1 of 1) 10 points Find g 0 ( x ) when g ( x ) = Z x π (6 + cos t ) dt . 1. g 0 ( x ) = 6 x + sin x 2. g 0 ( x ) = 6 - sin x 3. g 0 ( x ) = 6 + cos x correct 4. g 0 ( x ) = - sin x 5. g 0 ( x ) = 6 x - cos x Explanation: By the Fundamental 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 ) = 6 + cos x . keywords:

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Mcbiles, Emily – Homework 3 – Due: Sep 19 2006, 3:00 am – Inst: David Fonken 2 003 (part 1 of 1) 10 points If F ( x ) = Z x 0 2 e 12 sin 2 θ dθ , find the value of F 0 ( π/ 4). 1. F 0 ( π/ 4) = 2 e 6 correct 2. F 0 ( π/ 4) = 6 e 12 3. F 0 ( π/ 4) = 6 e 6 4. F 0 ( π/ 4) = 6 e 2 5. F 0 ( π/ 4) = 2 e 12 Explanation: By the Fundamental theorem of calculus, F 0 ( x ) = 2 e 12 sin 2 x . At x = π/ 4, therefore, F 0 ( π/ 4) = 2 e 6 since sin( π 4 ) = 1 2 . keywords: integral, FTC 004 (part 1 of 1) 10 points Evaluate the definite integral I = Z π/ 2 0 (3 cos x + sin x ) dx . 1. I = 0 2. I = 2 3. I = 3 4. I = 4 correct 5. I = 1 Explanation: By the Fundamental Theorem of Calculus, I = h F ( x ) i π/ 2 0 = F ( π 2 ) - F (0) for any anti-derivative F of f ( x ) = 3 cos x + sin x . Taking F ( x ) = 3 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 = 4 . keywords: integral, FTC, trig function 005 (part 1 of 1) 10 points Evaluate the definite integral I = Z 1 0 (6 + 9 x - 7 x 2 ) dx . Correct answer: 8 . 16667 . Explanation: By linearity, Z 1 0 (6 + 9 x - 7 x 2 ) dx = 6 Z 1 0 dx + 9 Z 1 0 x dx - 7 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 ,
Mcbiles, Emily – Homework 3 – Due: Sep 19 2006, 3:00 am – Inst: David Fonken 3 provided r 6 = - 1. Consequently, I = 49 6 = 8 . 16667 . keywords: definite integral, polynomial 006 (part 1 of 1) 10 points Determine the indefinite integral I = Z 3 + 4 x x dx .

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homework 3 solutions - Mcbiles Emily Homework 3 Due 3:00 am...

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