Exam #2 - Version 073 Exam 2 Fouli(58395 This print-out...

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Version 073 – Exam 2 – Fouli – (58395) 1 This print-out should have 18 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 ) = 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 )
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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. 1. rate = 52 gal/min 2. rate = 50 gal/min correct 3. rate = 54 gal/min 4. rate = 53 gal/min 5. rate = 45 gal/min Explanation: By the Chain Rule, V ( t ) = 2000 20 parenleftBig 1 1 40 t parenrightBig = 100 parenleftBig 1 1 40 t parenrightBig , the negative sign indicating that the volume is decreasing. Consequently, after 20 minutes the water is draining from the tank at a rate = 100 parenleftbigg 1 20 40 parenrightbigg = 50 gal / min . 006 10.0 points A 10 foot ladder is leaning against a wall. If the foot of the ladder is sliding away from the wall at a rate of 5 ft/sec, at what speed is the top of the ladder falling when the foot of
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Version 073 – Exam 2 – Fouli – (58395) 3 the ladder is 6 feet away from the base of the wall?
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