M408L hwk1 solutions

# M408L hwk1 solutions - Pham Quoc – Homework 1 – Due...

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Unformatted text preview: Pham, Quoc – Homework 1 – Due: Jan 24 2007, 3:00 am – Inst: Eric Katerman 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 Determine the value of f (2) when f ( t ) = 6 t + 1 , f (- 1) = 3 . Correct answer: 15 . Explanation: The most general anti-derivative of f has the form F ( t ) = 3 t 2 + t + C , where C is an arbitrary constant whose value is determined by the condition F (- 1) = 3. For then F (- 1) = 3- 1 + C = 3 , i . e ., C = 1 . Consequently, f ( t ) = 3 t 2 + t + 1 , and so at t = 2 , f (2) = 15 . keywords: antiderivatives, affine function, particular values 002 (part 1 of 1) 10 points Consider the following functions: ( A ) F 1 ( x ) = cos2 x 4 , ( B ) F 2 ( x ) = sin 2 x, ( 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 only 3. none of them 4. all of them 5. F 1 and F 2 only 6. F 3 only correct 7. F 2 and F 3 only 8. F 1 only Explanation: By trig identities, cos2 x = 2cos 2 x- 1 = 1- 2sin 2 x, while sin2 x = 2sin 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 ) Not anti-derivative. ( C ) Anti-derivative. keywords: antiderivative, trig function, dou- ble angle formula, T/F, 003 (part 1 of 1) 10 points Find f ( t ) when f ( t ) = 2 3 cos 1 3 t- 8sin 2 3 t and f ( π 2 ) = 9. 1. f ( t ) = 10cos 1 3 t- 12sin 2 3 t + 10 2. f ( t ) = 2sin 1 3 t + 4cos 2 3 t + 6 Pham, Quoc – Homework 1 – Due: Jan 24 2007, 3:00 am – Inst: Eric Katerman 2 3. f ( t ) = 2sin 1 3 t + 12cos 2 3 t + 2 correct 4. f ( t ) = 2cos 1 3 t + 12sin 2 3 t + 2 5. f ( t ) = 2cos 1 3 t + 4sin 2 3 t + 6 6. f ( t ) = 10sin 1 3 t- 12cos 2 3 t + 10 Explanation: The function f must have the form f ( t ) = 2sin 1 3 t + 12cos 2 3 t + C where the constant C is determined by the condition f ‡ π 2 · = 2sin π 6 + 12cos π 3 + C = 9 . But by known trig values sin π 6 = cos π 3 = 1 2 , so 7 + C = 9. Consequently, f ( t ) = 2sin 1 3 t + 12cos 2 3 t + 2 . keywords: antiderivatives, trigonometric functions 004 (part 1 of 1) 10 points Find the value of f (- 1) when f 00 ( t ) = 18 t + 3 and f (1) = 4 , f (1) = 1 . Correct answer: 11 . Explanation: The most general anti-derivative of f 00 has the form f ( t ) = 9 t 2 + 3 t + C where C is an arbitrary constant. But if f (1) = 4, then f (1) = 9 + 3 + C = 4 , i . e . C =- 8 . Thus f ( t ) = 9 t 2 + 3 t- 8 , from which it follows that f ( t ) = 3 t 3 + 3 2 t 2- 8 t + D , where the constant D is determined by the condition f (1) = 3 + 3 2- 8 + D = 1 , i . e . D = 9 2 ....
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M408L hwk1 solutions - Pham Quoc – Homework 1 – Due...

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