HW4-solutions - merino(aem2588 – HW4 – chen –(55405 1...

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Unformatted text preview: merino (aem2588) – HW4 – chen – (55405) 1 This print-out should have 20 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 when f ( x ) = sin x 2 + cos x . 1. f ′ ( x ) = 2 cos x + sin 2 x (2 + cos x ) 2 2. f ′ ( x ) = 2 cos x + 1 (2 + cos x ) 2 correct 3. f ′ ( x ) = 2 cos x- 1 (2 + cos x ) 2 4. f ′ ( x ) = 2 cos x- 1 2 + cos x 5. f ′ ( x ) = 2 cos x + 1 2 + cos x Explanation: By the Quotient Rule, f ′ ( x ) = cos x (2 + cos x ) + sin x sin x (2 + cos x ) 2 = 2 cos x + (cos 2 x + sin 2 x ) (2 + cos x ) 2 . But cos 2 x + sin 2 x = 1, so f ′ ( x ) = 2 cos x + 1 (2 + cos x ) 2 . 002 10.0 points Find dy dx when x 3 y 3- y = x . 1. dy dx = 1- 3 x 3 y 3 3 x 3 y 2- 2 2. dy dx = 1 + 3 x 2 y 3 3 x 3 y 2- 2 3. dy dx = 1- 3 x 2 y 3 3 x 3 y 2- 1 correct 4. dy dx = 1- 2 x 3 y 3 2 x 3 y 2- 1 5. dy dx = 1- 3 x 3 y 2 3 x 2 y 2- 1 Explanation: Differentiating x 3 y 3- y = x implicitly with respect to x we see that 3 x 3 y 2 dy dx + 3 y 3 x 2- dy dx = 1 . Thus 3 x 3 y 2 dy dx- dy dx = 1- 3 x 2 y 3 and so dy dx = 1- 3 x 2 y 3 3 x 3 y 2- 1 . 003 10.0 points Determine f ′ ( x ) when f ( x ) =- 4 x 7- 3 x 3- 9 π. 1. f ′ ( x ) =- 28 x 6- 9 x 2- 9 πx 2. f ′ ( x ) =- 28 x 6- 9 x 2- 9 π 3. f ′ ( x ) =- 4 x 7- 9 x 2- 9 4. f ′ ( x ) =- 28 x 6- 9 x 2 correct 5. f ′ ( x ) =- 28 x 6- 3 x 3 Explanation: By linearity of differentiation and the rule ( x n ) ′ = n · x n − 1 , merino (aem2588) – HW4 – chen – (55405) 2 we see that f ′ ( x ) =- 28 x 6- 9 x 2 . 004 10.0 points Find the derivative of f when f ( x ) = cos(sin x ) . 1. f ′ ( x ) = sin x cos(cos x ) 2. f ′ ( x ) =- cos x sin(cos x ) 3. f ′ ( x ) = cos x sin(sin x ) 4. f ′ ( x ) =- sin x cos(sin x ) 5. f ′ ( x ) =- cos x sin(sin x ) correct 6. f ′ ( x ) = sin x cos(sin x ) Explanation: Using the Chain Rule we see that f ′ ( x ) = parenleftBig- sin(sin x ) parenrightBig (cos x ) =- cos x sin(sin x ) . 005 10.0 points Find dy/dx when 4 x 2 + 5 y 2 = 6 . 1. dy dx = 4 x 5 y 2. dy dx =- 4 xy 3. dy dx =- 4 x y 4. dy dx = x 5 y 5. dy dx =- 4 x 5 y correct 6. dy dx = 5 xy Explanation: Diferentiating 4 x 2 + 5 y 2 = 6 implicitly with respect to x we see that 8 x + 10 y dy dx = 0 . Consequently, dy dx =- 8 x 10 y =- 4 x 5 y ....
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This note was uploaded on 02/23/2011 for the course MATH 408C taught by Professor Knopf during the Spring '10 term at University of Texas.

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HW4-solutions - merino(aem2588 – HW4 – chen –(55405 1...

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