HW8-solutions

# HW8-solutions - wei(jw35975 HW8 milburn(54685 This...

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wei (jw35975) – HW8 – milburn – (54685) 1 This print-out should have 24 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 ( x ) = 3 x x - 3 x . 1. f ( x ) = 18 x ( x 2 - 3) 2 2. f ( x ) = - 3 1 - 3 x 2 3. f ( x ) = 3 1 + 3 x 2 4. f ( x ) = 18 x x 2 - 3 5. f ( x ) = - 18 x ( x 2 - 3) 2 correct 6. f ( x ) = - 18 x x - 3 Explanation: It’s best to simplify the function before dif- ferentiating: f ( x ) = 3 x 2 x 2 - 3 . Then by the Quotient Rule, f ( x ) = 6 x ( x 2 - 3) - 2 x (3 x 2 ) ( x 2 - 3) 2 Consequently, f ( x ) = - 18 x ( x 2 - 3) 2 . 002 10.0points Find the derivative of f when f ( x ) = x ( x - 2) . 1. f ( x ) = 2 x - 2 2 x 2. f ( x ) = 2 x + 2 x x 3. f ( x ) = 3 x + 2 x x 4. f ( x ) = 2 x - 2 x x 5. f ( x ) = 3 x + 2 2 x 6. f ( x ) = 3 x - 2 2 x correct Explanation: By the Product Rule f ( x ) = x - 2 2 x + x . After simplification this becomes f ( x ) = x - 2 + 2 x 2 x = 3 x - 2 2 x . 003 10.0points Find the value of f (4) when f ( x ) = 4 3 x 3 / 2 + 2 x 1 / 2 . 1. f (4) = 11 2 2. f (4) = 4 3. f (4) = 5 4. f (4) = 9 2 correct 5. f (4) = 6 Explanation:

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wei (jw35975) – HW8 – milburn – (54685) 2 Since d dx x r = rx r 1 , we see that f ( x ) = 2 x 1 / 2 + x 1 / 2 . At x = 4, therefore, f (4) = 9 2 . 004 10.0points Find the derivative of g ( x ) = parenleftbigg x + 2 x + 3 parenrightbigg (2 x - 3) . 1. g ( x ) = 2 x 2 + 12 x + 9 ( x + 3) 2 correct 2. g ( x ) = x 2 + 12 x - 9 ( x + 3) 2 3. g ( x ) = x 2 - 12 x + 9 x + 3 4. g ( x ) = 2 x 2 - 12 x - 9 ( x + 3) 2 5. g ( x ) = 2 x 2 - 12 x - 9 x + 3 6. g ( x ) = 2 x 2 + 12 x + 9 x + 3 Explanation: By the Quotient and Product Rules we see that g ( x ) = 2 braceleftbigg x + 2 x + 3 bracerightbigg + (2 x - 3) braceleftbigg ( x + 3) - ( x + 2) ( x + 3) 2 bracerightbigg = 2 braceleftbigg x + 2 x + 3 bracerightbigg + braceleftbigg 2 x - 3 ( x + 3) 2 bracerightbigg = 2( x + 2)( x + 3) + (2 x - 3) ( x + 3) 2 . But 2( x + 2)( x + 3) + (2 x - 3) = 2 x 2 + 12 x + 9 . Consequently g ( x ) = 2 x 2 + 12 x + 9 ( x + 3) 2 . 005 10.0points Apply the Product Rule twice to determine the derivative of f when f ( x ) = ( x + 1)( x - 2)( x - 3) . 1. f ( x ) = 3 x 2 + 4 x - 5 2. f ( x ) = 3 x 2 - 8 x + 1 correct 3. f ( x ) = 3 x 2 + 8 x + 1 4. f ( x ) = 3 x 2 - 4 x - 5 5. f ( x ) = ( x + 1)( x - 2) 6. f ( x ) = ( x - 2)( x - 3) Explanation: Applying the Product Rule to f once we see that f ( x ) = ( x - 2)( x - 3)+( x +1) d dx ( x - 2)( x - 3) . Applying the Product a second time, but now to the second term, we see that d dx ( x - 2)( x - 3) = ( x - 3)+( x - 2) = 2 x - 5 . Thus f ( x ) = ( x - 2)( x - 3) + ( x + 1)(2 x - 5) = ( x 2 - 5 x + 6) + (2 x 2 - 3 x - 5) . Consequently, f ( x ) = 3 x 2 - 8 x + 1 .
wei (jw35975) – HW8 – milburn – (54685) 3 006 10.0points Find the derivative of F when F ( x ) = f ( x ) - 3 x f ( x ) + 3 x and f is a differentiable function.

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