HW 1 - Portillo Tom Homework 1 Due Sep 5 2006 3:00 am Inst...

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Portillo, Tom – Homework 1 – Due: Sep 5 2006, 3:00 am – Inst: David Benzvi 1 This print-out should have 15 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 Stewart Section 4.10, Example 2, page 301 Find all functions g such that g 0 ( x ) = 3 x 2 + x + 4 x . 1. g ( x ) = 2 x ( 3 x 2 + x - 4 ) + C 2. g ( x ) = x 3 5 x 2 + 1 3 x + 4 + C 3. g ( x ) = 2 x 3 5 x 2 + 1 3 x + 4 + C cor- rect 4. g ( x ) = x ( 3 x 2 + x + 4 ) + C 5. g ( x ) = 2 x ( 3 x 2 + x + 4 ) + C 6. g ( x ) = 2 x 3 5 x 2 + 1 3 x - 4 + C Explanation: After division g 0 ( x ) = 3 x 3 / 2 + x 1 / 2 + 4 x - 1 / 2 , so we can now find an antiderivative of each term separately. But d dx ax r r = ax r - 1 for all a and all r 6 = 0. Thus 6 5 x 5 / 2 + 2 3 x 3 / 2 + 8 x 1 / 2 = 2 x 3 5 x 2 + 1 3 x + 4 is an antiderivative of g 0 . Consequently, g ( x ) = 2 x 3 5 x 2 + 1 3 x + 4 + C with C an arbitrary constant. keywords: antiderivative, power functions 002 (part 1 of 1) 10 points Find the most general antiderivative, F , of the function f ( x ) = 6 x 2 - 16 x + 8 . 1. F ( x ) = 2 x 3 - 8 x 2 + 8 x 2. F ( x ) = 2 x 3 - 8 x 2 + 8 x + C correct 3. F ( x ) = 6 x 3 - 16 x 2 + 8 x + C 4. F ( x ) = 2 x 3 + 8 x 2 + 8 x + C 5. F ( x ) = 2 x 3 + 8 x 2 + 8 x Explanation: Since d dx x r = rx r - 1 , the most general anti-derivative of f is the function F ( x ) = 6 x 3 3 - 16 x 2 2 + 8 x + C with C an arbitrary constant. Consequently, F ( x ) = 2 x 3 - 8 x 2 + 8 x + C . keywords: antiderivative, polynomial 003 (part 1 of 1) 10 points Consider the following functions: ( A ) F 1 ( x ) = - cos 2 x 4 ,
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Portillo, Tom – Homework 1 – Due: Sep 5 2006, 3:00 am – Inst: David Benzvi 2 ( B ) F 2 ( x ) = sin 2 x 2 , ( C ) F 3 ( x ) = - cos 2 x 2 . Which are anti-derivatives of f ( x ) = sin x cos x ? 1. F 2 and F 3 only 2. F 2 only 3. F 3 only 4. F 1 and F 2 only 5. F 1 and F 3 only 6. all of them correct 7. F 1 only 8. none of them Explanation: By trig identities, cos 2 x = 2 cos 2 x - 1 = 1 - 2 sin 2 x , while sin 2 x = 2 sin x cos x . But d dx sin x = cos x, d dx cos x = - sin x . Consequently, by the Chain Rule, ( A ) Anti-derivative. ( B ) Anti-derivative. ( C ) Anti-derivative. keywords: antiderivative, trig function, dou- ble angle formula, T/F, 004 (part 1 of 1) 10 points Find f ( t ) when f 0 ( t ) = 5 3 cos 1 3 t - 2 sin 2 3 t and f ( π 2 ) = 7. 1. f ( t ) = 5 cos 1 3 t + 3 sin 2 3 t + 3 2. f ( t ) = 5 cos 1 3 t + sin 2 3 t + 4 3. f ( t ) = 5 sin 1 3 t + cos 2 3 t + 4 4. f ( t ) = 7 sin 1 3 t - 3 cos 2 3 t + 5 5. f ( t ) = 7 cos 1 3 t - 3 sin 2 3 t + 5 6. f ( t ) = 5 sin 1 3 t + 3 cos 2 3 t + 3 correct Explanation: The function f must have the form f ( t ) = 5 sin 1 3 t + 3 cos 2 3 t + C where the constant C is determined by the condition f π 2 · = 5 sin π 6 + 3 cos π 3 + C = 7 .
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