homework 1 solutions - Mcbiles, Emily Homework 1 Due: Sep 5...

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Mcbiles, Emily – Homework 1 – Due: Sep 5 2006, 3:00 am – Inst: David Fonken 1 This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page – fnd all choices be±ore answering. The due time is Central time. 001 (part 1 o± 1) 10 points Stewart Section 4.10, Example 2, page 301 Find all ±unctions g such that g 0 ( x ) = x 2 + 3 x + 1 x . 1. g ( x ) = x µ 1 5 x 2 + x + 1 + C 2. g ( x ) = 2 x µ 1 5 x 2 + x + 1 + C cor- rect 3. g ( x ) = x ( x 2 + 3 x + 1 ) + C 4. g ( x ) = 2 x ( x 2 + 3 x + 1 ) + C 5. g ( x ) = 2 x µ 1 5 x 2 + x - 1 + C 6. g ( x ) = 2 x ( x 2 + 3 x - 1 ) + C Explanation: A±ter division g 0 ( x ) = x 3 / 2 + 3 x 1 / 2 + x - 1 / 2 , so we can now fnd an antiderivative o± each term separately. But d dx µ ax r r = ax r - 1 ±or all a and all r 6 = 0. Thus 2 5 x 5 / 2 + 2 x 3 / 2 + 2 x 1 / 2 = 2 x µ 1 5 x 2 + x + 1 is an antiderivative o± g 0 . Consequently, g ( x ) = 2 x µ 1 5 x 2 + x + 1 + C with C an arbitrary constant. keywords: antiderivative, power ±unctions 002 (part 1 o± 1) 10 points Find the most general antiderivative, F , o± the ±unction f ( x ) = 9 x 2 - 10 x + 8 . 1. F ( x ) = 9 x 3 - 10 x 2 + 8 x + C 2. F ( x ) = 3 x 3 - 5 x 2 + 8 x + C correct 3. F ( x ) = 3 x 3 - 5 x 2 + 8 x 4. F ( x ) = 3 x 3 + 5 x 2 + 8 x + C 5. F ( x ) = 3 x 3 + 5 x 2 + 8 x Explanation: Since d dx x r = rx r - 1 , the most general anti-derivative o± f is the ±unction F ( x ) = 9 µ x 3 3 - 10 µ x 2 2 + 8 x + C with C an arbitrary constant. Consequently, F ( x ) = 3 x 3 - 5 x 2 + 8 x + C . keywords: antiderivative, polynomial 003 (part 1 o± 1) 10 points Consider the ±ollowing ±unctions: ( A ) F 1 ( x ) = sin 2 x, ( B ) F 2 ( x ) = - cos 2 x 4 ,
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Mcbiles, Emily – Homework 1 – Due: Sep 5 2006, 3:00 am – Inst: David Fonken 2 ( C ) F 3 ( x ) = cos 2 x 2 . Which are anti-derivatives of f ( x ) = sin x cos x ? 1. F 1 and F 3 only 2. F 2 and F 3 only 3. F 2 only correct 4. F 1 only 5. F 3 only 6. none of them 7. F 1 and F 2 only 8. all 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 ) Not anti-derivative. ( B ) Anti-derivative. ( C ) Not 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 ) = 2 cos 1 3 t - 4 sin 2 3 t and f ( π 2 ) = 9. 1. f ( t ) = 6 sin 1 3 t + 2 cos 2 3 t + 5 2. f ( t ) = 6 cos 1 3 t + 2 sin 2 3 t + 5 3. f ( t ) = 6 sin 1 3 t + 6 cos 2 3 t + 3 correct 4. f ( t ) = 10 sin 1 3 t - 6 cos 2 3 t + 7 5. f ( t ) = 10 cos 1 3 t - 6 sin 2 3 t + 7 6. f ( t ) = 6 cos 1 3 t + 6 sin 2 3 t + 3 Explanation: The function f must have the form f ( t ) = 6 sin 1 3 t + 6 cos 2 3 t + C where the constant C is determined by the condition
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homework 1 solutions - Mcbiles, Emily Homework 1 Due: Sep 5...

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