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HW05-solutions

# HW05-solutions - white(taw933 HW05 ben-zvi(55600 This...

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white (taw933) – HW05 – 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 x -coordinates of all the points on the graph of f ( x ) = 2 x 3 + 2 x 2 2 x + 4 at which the tangent line is horizontal. 1. x -coord = 1 3 , 1 2. x -coord = 1 3 3. x -coords = 1 3 , 1 4. x -coord = 1 3 5. x -coords = 1 , 1 3 correct 6. x -coord = 1 Explanation: The tangent line will be horizontal at P ( x 0 , f ( x 0 )) when f ( x 0 ) = 0 . Now f ( x ) = 6 x 2 + 4 x 2 = 2(3 x 1)( x + 1) . Consequently, x 0 = 1 , 1 3 . 002 10.0points Find the value of f (1) when f ( x ) = 9 x 1 6 + 8 x 1 6 + 8 . Correct answer: 0 . 166667. Explanation: Since ( x r ) = r · x r 1 holds for all real numbers r , we see that f ( x ) = 3 2 x 5 6 4 3 x 7 6 . Hence f (1) = 1 6 . 003 10.0points Find the derivative of g ( x ) = parenleftbigg x + 3 x + 2 parenrightbigg (2 x 9) . 1. g ( x ) = x 2 8 x + 21 x + 2 2. g ( x ) = 2 x 2 8 x 21 ( x + 2) 2 3. g ( x ) = x 2 + 8 x 21 ( x + 2) 2 4. g ( x ) = 2 x 2 + 8 x + 21 x + 2 5. g ( x ) = 2 x 2 + 8 x + 21 ( x + 2) 2 correct 6. g ( x ) = 2 x 2 8 x 21 x + 2 Explanation: By the Quotient and Product Rules we see that g ( x ) = 2 braceleftbigg x + 3 x + 2 bracerightbigg + (2 x 9) braceleftbigg ( x + 2) ( x + 3) ( x + 2) 2 bracerightbigg = 2 braceleftbigg x + 3 x + 2 bracerightbigg + braceleftbigg 2 x 9 ( x + 2) 2 bracerightbigg = 2( x + 3)( x + 2) + (2 x 9) ( x + 2) 2 .

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white (taw933) – HW05 – ben-zvi – (55600) 2 But 2( x + 3)( x + 2) (2 x 9) = 2 x 2 + 8 x + 21 . Consequently g ( x ) = 2 x 2 + 8 x + 21 ( x + 2) 2 . 004 10.0points Find the derivative of f when f ( x ) = 4 x 3 + 3 x 2 + 4 π . 1. f ( x ) = 6 parenleftBig 2 x 5 + 1 x 3 parenrightBig 2. f ( x ) = 6 parenleftBig 2 x 5 1 x 3 parenrightBig correct 3. f ( x ) = 2 parenleftBig 2 x 4 + 1 x 2 parenrightBig 4. f ( x ) = 6 parenleftBig 1 2 x 5 x 3 parenrightBig 5. f ( x ) = 2 parenleftBig 2 x 4 1 x 2 parenrightBig 6. f ( x ) = 2 parenleftBig 1 2 x 5 x 3 parenrightBig 7. none of the other answers Explanation: Since d dx ( x r ) = rx r 1 holds for all r , we see that f ( x ) = 12 x 2 6 x 3 . Consequently, f ( x ) = 6 parenleftBig 2 x 5 1 x 3 parenrightBig . keywords: derivatives, negative powers 005 10.0points Find the derivative of f when f ( x ) = x x + 1 x . 1. f ( x ) = 1 1 1 x 2 2. f ( x ) = 2 x x + 1 3. f ( x ) = 2 x ( x 2 + 1) 2 4. f ( x ) = 2 x ( x 2 + 1) 2 correct 5. f ( x ) = 1 1 + 1 x 2 6. f ( x ) = 2 x x 2 + 1 Explanation: It’s best to simplify the function before dif- ferentiating; for then f ( x ) = x 2 x 2 + 1 .
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HW05-solutions - white(taw933 HW05 ben-zvi(55600 This...

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