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Unformatted text preview: rabbani (tar547) Review 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 lim x 5 x 2 x + 5 5 + 7 x 6 x 2 . 1. limit = 0 2. limit = 5 6 correct 3. none of the other answers 4. limit = 5. limit = 5 12 Explanation: Dividing the numerator and denominator by x 2 we see that 5 x 2 x + 5 5 + 7 x 6 x 2 = 5 1 x + 5 x 2 5 x 2 + 7 x 6 . On the other hand, lim x 1 x = lim x 1 x 2 = 0 . By Properties of limits, therefore, the limit = 5 6 . 002 10.0 points Find the derivative of f when f ( x ) = 5 tan4 x cos 3 4 x . 1. f ( x ) = 5 cos 4 x (1 3 cos 2 4 x ) 2. f ( x ) = 5 cos 4 x (3 sin 2 4 x 1) 3. f ( x ) = 20cos 4 x (1 3 cos 2 4 x ) 4. f ( x ) = 20cos 4 x (1 + 3 sin 2 4 x ) 5. f ( x ) = 20cos 4 x (1 3 sin 2 4 x ) correct Explanation: Using the fact that d dx tan x = 1 cos 2 x , d dx cos x = sin x, together with the Chain rule, we obtain f ( x ) = 20 cos 2 4 x cos 3 4 x 60tan 4 x cos 2 4 x sin 4 x. Consequently, f ( x ) = 20cos 4 x (1 3 sin 2 4 x ) . Notice that the problem slightly simpler if we observe that tan 4 x = sin 4 x cos 4 x , so that f ( x ) = 5 sin4 x cos 2 4 x, and then differentiate this function using the known derivatives of sin 4 x and cos 4 x . 003 10.0 points Find f ( x ) when f ( x ) = parenleftBig x 5 x 2 + 1 parenrightBig 2 . 1. f ( x ) = x (1 5 x ) (5 x 2 + 1) 2 2. f ( x ) = 2(1 5 x 2 ) (5 x 2 + 1) 3 3. f ( x ) = 2 x (1 5 x 2 ) (5 x 2 + 1) 3 correct rabbani (tar547) Review 2 Fouli (58395) 2 4. f ( x ) = 2 x (1 5 x ) (5 x 2 + 1) 2 5. f ( x ) = 2(1 5 x 2 ) (5 x 2 + 1) 2 6. f ( x ) = x (1 5 x 2 ) (5 x 2 + 1) 3 Explanation: By the Power rule, f ( x ) = 2 parenleftBig x 5 x 2 + 1 parenrightBig d dx parenleftBig x 5 x 2 + 1 parenrightBig . But, by the Quotient rule, d dx parenleftBig x 5 x 2 + 1 parenrightBig = (5 x 2 + 1) 10 x 2 (5 x 2 + 1) 2 . Consequently, f ( x ) = 2 x (1 5 x 2 ) (5 x 2 + 1) 3 . 004 10.0 points Find the slope of the tangent line to the graph of 2 x 3 y 3 + xy = 0 at the point P ( 1 , 1). 1. slope = 5 4 correct 2. slope = 4 5 3. slope = 3 2 4. slope = 4 5 5. slope = 5 4 6. slope = 2 3 Explanation: Differentiating implicitly with respect to x we see that 6 x 2 3 y 2 dy dx + y + x dy dx = 0 . Consequently, dy dx = 6 x 2 + y 3 y 2 x . Hence at P ( 1 , 1) slope = dy dx vextendsingle vextendsingle vextendsingle P = 5 4 . 005 10.0 points Determine dy/dx when 2 cos x sin y = 5 ....
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 Fall '08
 JOUVE

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