HW6 (solutions) - myers(nmm698 HW06 Hamrick(54868 This...

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myers (nmm698) – HW06 – Hamrick – (54868) 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find the derivative of f ( x ) = 3 x sin 4 x + 3 4 cos 4 x. 1. f ( x ) = 12 x cos 4 x correct 2. f ( x ) = - 12 x cos 4 x 3. f ( x ) = 12 x cos 4 x - 6 sin 4 x 4. f ( x ) = 12 cos 4 x 5. f ( x ) = 12 x cos 4 x + 6 sin 4 x Explanation: Since d dx sin x = cos x, d dx cos x = - sin x, it follows that f ( x ) = 3 sin 4 x + 12 x cos 4 x - 3 sin 4 x. Consequently, f ( x ) = 12 x cos 4 x . 002 10.0 points Find f ( x ) when f ( x ) = parenleftBig x + 1 x - 1 parenrightBig 2 . 1. f ( x ) = - 4( x + 1) ( x - 1) 3 correct 2. f ( x ) = 6( x - 2) ( x + 1) 3 3. f ( x ) = - 6( x + 2) ( x - 1) 3 4. f ( x ) = 6( x - 1) ( x + 1) 3 5. f ( x ) = - 4( x + 2) ( x - 1) 3 6. f ( x ) = 4( x - 1) ( x + 1) 3 Explanation: By the Chain and Quotient Rules, f ( x ) = 2 parenleftBig x + 1 x - 1 parenrightBig ( x - 1) - ( x + 1) ( x - 1) 2 . Consequently, f ( x ) = - 4( x + 1) ( x - 1) 3 . 003 10.0 points Find f ( x ) when f ( x ) = 1 x 2 - 8 x . 1. f ( x ) = 4 - x ( x 2 - 8 x ) 3 / 2 correct 2. f ( x ) = x - 4 ( x 2 - 8 x ) 3 / 2 3. f ( x ) = 4 - x ( x 2 - 8 x ) 1 / 2 4. f ( x ) = 4 - x (8 x - x 2 ) 3 / 2 5. f ( x ) = x - 4 (8 x - x 2 ) 3 / 2 6. f ( x ) = x - 4 (8 x - x 2 ) 1 / 2 Explanation: By the Chain Rule, f ( x ) = - 1 2( x 2 - 8 x ) 3 / 2 (2 x - 8) .
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myers (nmm698) – HW06 – Hamrick – (54868) 2 Consequently, f ( x ) = 4 - x ( x 2 - 8 x ) 3 / 2 . 004 10.0 points Find f ( x ) when f ( x ) = 3 cos 2 x + 2 cos 2 x. 1. f ( x ) = - 8 cos 2 x 2. f ( x ) = 16 cos 2 x 3. f ( x ) = - 16 cos 2 x 4. f ( x ) = - 16 sin 2 x 5. f ( x ) = - 8 sin 2 x correct 6. f ( x ) = 8 sin 2 x Explanation: Differentiating once we see that f ( x ) = - 6 sin 2 x - 4 sin x cos x. Now 2 sin x cos x = sin 2 x. Consequently, f ( x ) = - 8 sin 2 x . 005 10.0 points Find the derivative of f when f ( x ) = cos(cos x ) . 1. f ( x ) = - cos x sin(cos x ) 2. f ( x ) = cos x sin(cos x ) 3. f ( x ) = sin x sin(cos x ) correct 4. f ( x ) = - sin x sin(cos x ) 5. f ( x ) = - cos x sin(sin x ) 6. f ( x ) = sin x sin(sin x ) Explanation: Using the Chain Rule we see that f ( x ) = parenleftBig - sin(cos x ) parenrightBig ( - sin x ) = sin x sin(cos x ) . 006 10.0 points Find the derivative of y when y = 10 sin x - 2 x cos x. 1. y = sin x - 6 parenleftBig cos x x parenrightBig 2. y = cos x + 6 parenleftBig sin x x parenrightBig 3. y = 5 sin x + 6 parenleftBig sin x x parenrightBig 4. y = sin x + 4
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