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Unformatted text preview: Version 073 – Exam 2 – Fouli – (58395) 1 This printout should have 18 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine f ′ ( x ) when f ( x ) = x − 2 √ x 2 + 1 . 1. f ′ ( x ) = 1 + 2 x ( x 2 + 1) 1 / 2 2. f ′ ( x ) = 1 − 2 x ( x 2 + 1) 1 / 2 3. f ′ ( x ) = 1 + 2 x ( x 2 + 1) 3 / 2 correct 4. f ′ ( x ) = 1 − 2 x ( x 2 + 1) 3 / 2 5. f ′ ( x ) = 2 x + 1 ( x 2 + 1) 1 / 2 6. f ′ ( x ) = 2 x − 1 ( x 2 + 1) 3 / 2 Explanation: By the Product and Chain Rules, f ′ ( x ) = 1 ( x 2 + 1) 1 / 2 − 2 x ( x − 2) 2( x 2 + 1) 3 / 2 = ( x 2 + 1) − x ( x − 2) ( x 2 + 1) 3 / 2 . Consequently, f ′ ( x ) = 1 + 2 x ( x 2 + 1) 3 / 2 . (Note: the Quotient Rule could have been used, but it’s simpler to use the Product Rule.) 002 10.0 points Determine f ′ ( x ) when f ( x ) = 2 tan 2 x − 3 sec 2 x . 1. f ′ ( x ) = 2 tan 2 sec x 2. f ′ ( x ) = − 2 tan 2 sec x 3. f ′ ( x ) = 2 sec 2 x tan x 4. f ′ ( x ) = − 2 sec 2 x tan x correct 5. f ′ ( x ) = 10 tan 2 sec x 6. f ′ ( x ) = 10 sec 2 x tan x Explanation: Since d dx sec x = sec x tan x, d dx tan x = sec 2 x, the Chain Rule ensures that f ′ ( x ) = 4 tan x sec 2 x − 6 sec 2 x tan x . Consequently, f ′ ( x ) = − 2 sec 2 x tan x . 003 10.0 points Determine dy/dx when y cos( x 2 ) = 4 . 1. dy dx = − 2 xy cot( x 2 ) 2. dy dx = − 2 xy sin( x 2 ) 3. dy dx = 2 xy cot( x 2 ) 4. dy dx = 2 xy cos( x 2 ) 5. dy dx = 2 xy tan( x 2 ) correct 6. dy dx = − 2 xy tan( x 2 ) Version 073 – Exam 2 – Fouli – (58395) 2 Explanation: After implicit differentiation with respect to x we see that − 2 xy sin( x 2 ) + y ′ cos( x 2 ) = 0 . Consequently, dy dx = 2 xy sin( x 2 ) cos( x 2 ) = 2 xy tan( x 2 ) . 004 10.0 points Find an equation for the tangent line to the curve 7 x 2 + xy + 2 y 2 = 10 at the point (1 , 1). 1. y = 9 x + 4 2. y = 5 x − 6 3. y = − 3 x + 4 correct 4. y = 3 x + 9 5. y = − 5 x + 6 6. y = − 9 x + 4 Explanation: Differentiating implicitly, we see that 7 x 2 + xy + 2 y 2 = 10 14 x + xy ′ + y · 1 + 4 yy ′ = 0 xy ′ + 4 yy ′ = − 14 x − y y ′ ( x + 4 y ) = − 14 x − y y ′ = − 14 x − y x + 4 y When x = 1 and y = 1, we have y ′ = − 14 − 1 1 + 4 = − 15 5 = − 3 so an equation of the tangent line is y − 1 = − 3 ( x − 1) y = − 3 x + 4 keywords: 005 10.0 points If a tank holds 2000 gallons of water, and the water can drain from the tank in 40 min utes, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as V = 2000 parenleftbigg 1 − t 40 parenrightbigg 2 . Find the rate at which water is draining from the tank after 20 minutes....
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This note was uploaded on 07/10/2011 for the course KIN 321M taught by Professor Jensen during the Spring '11 term at University of Texas.
 Spring '11
 Jensen

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