Review 2-solutions

Review 2-solutions - wtm369 – Review 2 – Helleloid...

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Unformatted text preview: wtm369 – Review 2 – Helleloid – (58250) 1 This print-out should have 19 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 ) = 2 x sin4 x + 1 2 cos 4 x . 1. f ′ ( x ) = 8 x cos 4 x correct 2. f ′ ( x ) =- 8 x cos 4 x 3. f ′ ( x ) = 8 cos 4 x 4. f ′ ( x ) = 8 x cos 4 x- 4 sin 4 x 5. f ′ ( x ) = 8 x cos 4 x + 4 sin 4 x Explanation: Since d dx sin x = cos x, d dx cos x =- sin x, it follows that f ′ ( x ) = 2 sin4 x + 8 x cos4 x- 2 sin4 x. Consequently, f ′ ( x ) = 8 x cos4 x . 002 10.0 points Determine the derivative of f ( x ) = x radicalbig x 2 + 1 . 1. f ′ ( x ) = x 2 + 2 √ x 2 + 1 2. f ′ ( x ) = x 2- 2 √ x 2 + 1 3. f ′ ( x ) = 1- 2 x 2 √ x 2 + 1 4. f ′ ( x ) = 2 x 2- 1 √ x 2 + 1 5. f ′ ( x ) = 2 x 2 + 1 √ x 2 + 1 correct 6. f ′ ( x ) = 2- x 2 √ x 2 + 1 Explanation: By the Product and Chain Rules, f ′ ( x ) = radicalbig x 2 + 1 + x 2 √ x 2 + 1 = ( x 2 + 1) + x 2 √ x 2 + 1 . Consequently, f ′ ( x ) = 2 x 2 + 1 √ x 2 + 1 . keywords: 003 10.0 points Determine f ′′ ( x ) when f ( x ) = sin 2 x + 2 cos x . 1. f ′′ ( x ) = 2 cos2 x- 2 cos x correct 2. f ′′ ( x ) = 2 cos2 x + 2 cos x 3. f ′′ ( x ) = cos 2 x + 2 sin x 4. f ′′ ( x ) = 2 sin2 x- 2 cos x 5. f ′′ ( x ) = cos 2 x- 2 cos x 6. f ′′ ( x ) = sin 2 x + 2 sin x Explanation: After differentiation, f ′ ( x ) = 2 sin x cos x- 2 sin x . wtm369 – Review 2 – Helleloid – (58250) 2 But by trig identities, 2 sin x cos x = sin 2 x . Thus f ′ ( x ) = sin 2 x- 2 sin x . Consequently, after a second differentiation, f ′′ ( x ) = 2 cos 2 x- 2 cos x . 004 10.0 points Find an equation for the tangent line to the curve x 2 + xy + y 2 = 3 at the point (1 , 1). 1. y =- 2 x + 2 2. y = 2 x- 3 3. y = 6 x + 2 4. y =- 6 x + 2 5. y = x + 5 6. y =- x + 2 correct Explanation: Differentiating implicitly, we see that x 2 + xy + y 2 = 3 2 x + xy ′ + y · 1 + 2 yy ′ = 0 xy ′ + 2 yy ′ =- 2 x- y y ′ ( x + 2 y ) =- 2 x- y y ′ =- 2 x- y x + 2 y When x = 1 and y = 1, we have y ′ =- 2- 1 1 + 2 =- 3 3 =- 1 so an equation of the tangent line is y- 1 =- 1 ( x- 1) y =- x + 2 keywords: 005 10.0 points TI has discovered that the total cost (in dollars) of producing x calculators per day is given by C ( x ) = 200 + 70 x- x 2 9 . Find the Marginal Cost to TI at a production level of x calculators. 1. Marginal Cost = C ( x ) x 2. Marginal Cost = 70 + 1 9 x 3. Marginal Cost = C ′ ( x ) = 70 + 2 9 x 4. Marginal Cost = 70- 1 9 x 5. Marginal Cost = C ( x + 1)- C ( x ) 6. Marginal Cost = C ′ ( x ) = 70- 2 9 x correct Explanation: The Marginal cost by definition is the derivative, C ′ , of the Cost function C . Thus Marginal Cost = 70- 2 9 x ....
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This note was uploaded on 12/04/2009 for the course M 408k taught by Professor Schultz during the Fall '08 term at University of Texas.

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Review 2-solutions - wtm369 – Review 2 – Helleloid...

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