HW06.pdf - young(toy68 HW06 villafuerte altu(53780 This...

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young (toy68) – HW06 – villafuerte altu – (53780) 1 This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Find an equation for the tangent to the graph of f at the point P (2 , f (2)) when f ( x ) = 5 1 3 x . 1. y = 3 5 x 11 5 correct 2. y = 3 x 7 3. y = 1 5 x 9 5 4. y + 3 5 x + 2 5 = 0 5. y + 1 5 x + 3 5 = 0 Explanation: If x = 2, then f (2) = 1, so we have to find an equation for the tangent line to the graph of f ( x ) = 5 1 3 x at the point (2 , 1). Now the Newtonian quotient for f at a general point ( x, f ( x )) is given by f ( x + h ) f ( x ) h . First let’s compute the numerator of the New- tonian Quotient: f ( x + h ) f ( x ) = 5 1 3( x + h ) 5 1 3 x = 5(1 3 x ) 5 { 1 3( x + h ) } (1 3 x ) { 1 3( x + h ) } = 15 h (1 3 h ) (1 3( x + h )) . Thus f ( x + h ) f ( x ) h = 15 (1 3 x ) (1 3( x + h )) . Hence f ( x ) = lim h 0 15 (1 3( x + h ))(1 3 x ) = 15 (1 3 x ) 2 . At x = 2, therefore, f (2) = 15 (1 6) 2 = 3 5 , so by the point slope formula an equation for the tangent line at (2 , 1) is y + 1 = 3 5 ( x 2) which after simplification becomes y = 3 5 x 11 5 . 002 10.0points Determine the derivative of f when f ( x ) = parenleftbigg 1 3 parenrightbigg 2 / 3 . 1. f ( x ) = 0 correct 2. f ( x ) = 2 9 x 1 / 3 3. f ( x ) = parenleftbigg 1 3 parenrightbigg x 1 / 3 4. f ( x ) = 2 3 parenleftbigg 1 3 parenrightbigg 1 / 3 5. f ( x ) does not exist Explanation: The derivative of any constant function is zero. Consequently, f ( x ) = 0 . 003 10.0points
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young (toy68) – HW06 – villafuerte altu – (53780) 2 Find the x -coordinate of all points on the graph of f ( x ) = x 3 x 2 x + 3 at which the tangent line is horizontal. 1. x -coords = 1 3 , 1 correct 2. x -coord = 1 3 3. x -coord = 1 4. x -coords = 1 3 , 1 5. x -coord = 1 3 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 ) = 3 x 2 2 x 1 = (3 x + 1)( x 1) . Consequently, x 0 = 1 3 , 1 . 004 10.0points Find the derivative of f ( x ) = x 2 x . 1. f ( x ) = x 2 2 x x 2. f ( x ) = x + 2 2 x x correct 3. f ( x ) = x 2 x 4. f ( x ) = x + 2 x x 5. f ( x ) = x + 2 2 x 6. f ( x ) = x 2 x x Explanation: Since d dx x = d dx x 1 / 2 = 1 2 x 1 / 2 = 1 2 x , while d dx 1 x = d dx x 1 / 2 = 1 2 x 3 / 2 = 1 2 x x . Thus f ( x ) = 1 2 x + 1 x x . Consequently, f ( x ) = x + 2 2 x x . 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 2 1) 2 correct 3. f ( x ) = 2 x x 1 4. f ( x ) = 1 1 1 x 2 5. f ( x ) = 2 x x 2 1 6. f ( x ) = 2 x ( x 2 1) 2 Explanation:
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young (toy68) – HW06 – villafuerte altu – (53780) 3 It’s best to simplify the function before dif- ferentiating; for then f ( x ) = x 2 x 2 1 . Thus by the Quotient Rule, f ( x ) = 2 x ( x 2 1) 2 x ( x 2 ) ( x 2 1) 2 Consequently, f ( x ) = 2 x ( x 2 1) 2 .
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