HW05-solutions

HW05-solutions - husain(aih243 – HW05 – Gilbert...

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Unformatted text preview: husain (aih243) – HW05 – Gilbert – (56215) 1 This print-out should have 25 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine f ′ ( x ) when f ( x ) = 2 sin3 x + 3 cos 2 x . 1. f ′ ( x ) = 6(cos 3 x + sin2 x ) 2. f ′ ( x ) =- 6(sin2 x + cos 3 x ) 3. f ′ ( x ) = 2 cos 3 x + 3 cos 2 x 4. f ′ ( x ) = 6(cos 3 x- sin2 x ) correct 5. f ′ ( x ) = 2 cos 3 x- 3 sin2 x 6. f ′ ( x ) =- (3 sin2 x + 2 cos3 x ) Explanation: Since d dx sin x = cos x , d dx cos x =- sin x , the Chain Rule ensures that f ′ ( x ) = 6(cos 3 x- sin2 x ) . 002 10.0 points Find f ′ ( x ) when f ( x ) = parenleftBig x- 2 x + 1 parenrightBig 2 . 1. f ′ ( x ) =- 6( x + 2) ( x- 1) 3 2. f ′ ( x ) =- 4( x- 2) ( x + 1) 3 3. f ′ ( x ) = 6( x + 1) ( x- 1) 3 4. f ′ ( x ) =- 4( x + 1) ( x- 1) 3 5. f ′ ( x ) = 4( x- 1) ( x + 1) 3 6. f ′ ( x ) = 6( x- 2) ( x + 1) 3 correct Explanation: By the Chain and Quotient Rules, f ′ ( x ) = 2 parenleftBig x- 2 x + 1 parenrightBig ( x + 1)- ( x- 2) ( x + 1) 2 . Consequently, f ′ ( x ) = 6( x- 2) ( x + 1) 3 . 003 10.0 points Determine f ′ ( x ) when f ( x ) = x- 1 √ x 2- 2 . 1. f ′ ( x ) = 2- x ( x 2- 2) 3 / 2 2. f ′ ( x ) = x- 2 ( x 2- 2) 3 / 2 correct 3. f ′ ( x ) = 2 + x ( x 2- 2) 1 / 2 4. f ′ ( x ) = x- 2 ( x 2- 2) 1 / 2 5. f ′ ( x ) = 2 + x ( x 2- 2) 3 / 2 6. f ′ ( x ) = 2- x ( x 2- 2) 1 / 2 Explanation: By the Product and Chain Rules, f ′ ( x ) = 1 ( x 2- 2) 1 / 2- 2 x ( x- 1) 2( x 2- 2) 3 / 2 = ( x 2- 2)- x ( x- 1) ( x 2- 2) 3 / 2 . husain (aih243) – HW05 – Gilbert – (56215) 2 Consequently, f ′ ( x ) = x- 2 ( x 2- 2) 3 / 2 . (Note: the Quotient Rule could have been used, but it’s simpler to use the Product Rule.) keywords: derivative, square root, chain rule, product rule 004 10.0 points Determine the derivative of f ( x ) = ( x- 3) √ x- 2 . 1. f ′ ( x ) = 3 x + 7 x- 2 2. f ′ ( x ) = 3 x + 7 2( x- 2) 3. f ′ ( x ) = 3 x- 7 2 √ x- 2 correct 4. f ′ ( x ) = 3 x + 7 √ x- 2 5. f ′ ( x ) = 3 x- 7 √ x- 2 6. f ′ ( x ) = 3 x- 7 2( x- 2) Explanation: By the Product and Chain Rules, f ′ ( x ) = 1 2 √ x- 2 ( x- 3) + √ x- 2 = x- 3 + 2( x- 2) 2 √ x- 2 . Consequently, f ′ ( x ) = 3 x- 7 2 √ x- 2 . keywords: Product Rule, Chain Rule, Power Rule, square root function, 005 10.0 points Find the first derivative of f when f ( x ) = 2 cos2 x- sin 2 x . 1. f ′ ( x ) =- 10 sin2 x 2. f ′ ( x ) =- 5 cos 2 x 3. f ′ ( x ) =- 5 sin2 x correct 4. f ′ ( x ) = 10 sin2 x 5. f ′ ( x ) = 5 sin2 x 6. f ′ ( x ) =- 10 cos2 x Explanation: By the Chain Rule we see that f ′ ( x ) =- 4 sin2 x- 2 sin x cos x ....
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HW05-solutions - husain(aih243 – HW05 – Gilbert...

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