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Unformatted text preview: Version 121 Exam 2 Fouli (58320) 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 + 1 x 2 2 . 1. f ( x ) = 2 x ( x 2 2) 3 / 2 2. f ( x ) = 2 + x ( x 2 2) 3 / 2 correct 3. f ( x ) = 2 x ( x 2 2) 1 / 2 4. f ( x ) = 2 + x ( x 2 2) 1 / 2 5. f ( x ) = 2 + x ( x 2 2) 1 / 2 6. f ( x ) = 2 + x ( x 2 2) 3 / 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 . Consequently, f ( x ) = 2 + x ( x 2 2) 3 / 2 . (Note: the Quotient Rule could have been used, but its simpler to use the Product Rule.) 002 10.0 points Determine f ( x ) when f ( x ) = 3 tan 2 x sec 2 x . 1. f ( x ) = 4 tan 2 sec x 2. f ( x ) = 4 sec 2 x tan x correct 3. f ( x ) = 8 tan 2 sec x 4. f ( x ) = 4 tan 2 sec x 5. f ( x ) = 4 sec 2 x tan x 6. f ( x ) = 8 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 ) = 6 tan x sec 2 x 2 sec 2 x tan x . Consequently, f ( x ) = 4 sec 2 x tan x . 003 10.0 points Determine dy/dx when y cos( x 2 ) = 2 . 1. dy dx = 2 xy cos( x 2 ) 2. dy dx = 2 xy cot( x 2 ) 3. dy dx = 2 xy cot( x 2 ) 4. dy dx = 2 xy sin( x 2 ) 5. dy dx = 2 xy tan( x 2 ) correct 6. dy dx = 2 xy tan( x 2 ) Version 121 Exam 2 Fouli (58320) 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 4 x 2 + xy + y 2 = 6 at the point (1 , 1). 1. y = 3 x + 8 2. y = 9 x + 4 3. y = 4 x 5 4. y = 9 x + 4 5. y = 3 x + 4 correct 6. y = 4 x + 5 Explanation: Differentiating implicitly, we see that 4 x 2 + xy + y 2 = 6 8 x + xy + y 1 + 2 yy = 0 xy + 2 yy = 8 x y y ( x + 2 y ) = 8 x y y = 8 x y x + 2 y When x = 1 and y = 1, we have y = 8 1 1 + 2 = 9 3 = 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 Torricellis 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 09/09/2009 for the course M 408L taught by Professor Gilbert during the Spring '09 term at University of TexasTyler.
 Spring '09
 GILBERT

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