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HW14-solutions

# HW14-solutions - white(taw933 HW14 ben-zvi(55600 This...

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white (taw933) – HW14 – ben-zvi – (55600) 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 10.0points Find the derivative of f when f ( θ ) = ln (cos 3 θ ) . 1. f ( θ ) = - 1 sin 3 θ 2. f ( θ ) = - 3 cot 3 θ 3. f ( θ ) = 3 cos 3 θ 4. f ( θ ) = cot 3 θ 5. f ( θ ) = - 3 tan 3 θ correct 6. f ( θ ) = 3 tan 3 θ Explanation: By the Chain Rule, f ( θ ) = 1 cos(3 θ ) d (cos 3 θ ) = - 3 sin 3 θ cos 3 θ . Consequently, f ( θ ) = - 3 tan 3 θ . 002 10.0points Find the slope of the line tangent to the graph of ln( xy ) - 2 x = 0 at the point where x = 1. 1. slope = 1 2 e 2 2. slope = e 2 correct 3. slope = - 1 2 e 2 4. slope = e 2 5. slope = - e 2 6. slope = - 1 2 e 2 Explanation: Differentiating implicitly with respect to x we see that 1 xy parenleftBig y + x dy dx parenrightBig - 2 = 0 , in which case dy dx = - y (1 - 2 x ) x = - e 2 x (1 - 2 x ) x 2 because, by exponentiation, y = e 2 x x . Consequently, at x = 1, slope = dy dx vextendsingle vextendsingle vextendsingle x =1 = e 2 . 003 10.0points Find the derivative of f ( t ) = 1 + ln t 4 - ln t . 1. f ( t ) = 5 t (4 - ln t ) 2 correct 2. f ( t ) = 4 t (1 + ln t ) 2 3. f ( t ) = - 4 ln t t (1 + ln t ) 2 4. f ( t ) = - 5 t (4 - ln t ) 2 5. f ( t ) = 4 ln t (1 + ln t ) 2 6. f ( t ) = - 5 (4 - ln t ) 2

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white (taw933) – HW14 – ben-zvi – (55600) 2 Explanation: By the Quotient Rule, f ( t ) = (4 - ln t )(1 /t ) + (1 + ln t )(1 /t ) (4 - ln t ) 2 = (4 - ln t ) + (1 + ln t ) t (4 - ln t ) 2 . Consequently, f ( t ) = 5 t (4 - ln t ) 2 . 004 10.0points Find the derivative of f when f ( x ) = 2 ln( x - radicalbig x 2 - 3) , ( x > 3) . 1. f ( x ) = - 4 x 2 - 3 2. f ( x ) = 4 x 2 - 3 3. f ( x ) = 2 x 2 - 3 4. f ( x ) = - 2 x 2 - 3 correct 5. f ( x ) = 1 x 2 - 3 6. f ( x ) = - 1 x 2 - 3 Explanation: By the Chain Rule f ( x ) = 2 x - x 2 - 3 parenleftBig 1 - x x 2 - 3 parenrightBig = - 2 x 2 - 3 . 005 10.0points Determine f ( x ) when f ( x ) = e (3ln( x 5 )) . 1. f ( x ) = 1 x e 3ln( x 5 ) 2. f ( x ) = 15 x 14 correct 3. f ( x ) = 15(ln x ) e 3ln( x 5 ) 4. f ( x ) = e 15 /x 5. f ( x ) = 14 x 15 6. f ( x ) = 3 x 2 e 3ln( x 5 ) Explanation: Since r ln x = ln x r , e ln x = x , we see that f ( x ) = e (ln x 15 ) = x 15 . Consequently, f ( x ) = 15 x 14 . 006 10.0points Find the derivative of f ( x ) = ln radicalBigg 1 + 2 x 2 1 - 2 x 2 .
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HW14-solutions - white(taw933 HW14 ben-zvi(55600 This...

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