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Unformatted text preview: Mahon, Kevin – Homework 5 – Due: Sep 29 2005, 3:00 am – Inst: Edward Odell 1 This printout should have 22 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points If f is a function defined on ( 2 , 2) whose graph is 1 2 1 2 1 2 1 2 which of the following is the graph of its derivative f ? 1. 1 2 1 2 1 2 1 2 2. 1 2 1 2 1 2 1 2 3. 1 2 1 2 1 2 1 2 4. 1 2 1 2 1 2 1 2 5. 1 2 1 2 1 2 1 2 6. 1 2 1 2 1 2 1 2 correct Explanation: Since the graph on ( 2 , 2) consists of straight lines joined at x = 1 and x = 1, the derivative of f will exist at all points in ( 2 , 2) except x = 1 and x = 1, eliminating the answer whose graph contain filled dots at x = 1 and x = 1. On the other hand, the graph of f (i) has slope 2 on ( 2 , 1), (ii) has slope 2 on ( 1 , 1), and Mahon, Kevin – Homework 5 – Due: Sep 29 2005, 3:00 am – Inst: Edward Odell 2 (iii) has slope 2 on (1 , 2). Consequently, the graph of f consists of the horizontal lines and ‘holes’ in 1 2 1 2 1 2 1 2 keywords: Stewart5e, graph, derivative, piecewise linear function 002 (part 1 of 2) 10 points Compute the lefthand derivative f ( a ) = lim h → f ( a + h ) f ( a ) h and righthand derivative f + ( a ) = lim h → + f ( a + h ) f ( a ) h of f at x = 5 when f ( x ) = 6 x, x < 5 , 1 6 x , x ≥ 5 . 1. f (5) = 1 , f + (5) = 1 2. f (5) = 1 , f + (5) = 1 correct 3. f (5) = 2 , f + (5) = 2 4. f (5) = 2 , f + (5) = 2 5. f (5) = 1 , f + (5) = 1 6. f (5) = 5 , f + (5) = 5 7. f (5) = 5 , f + (5) = 5 Explanation: Since f (5) = 1, the definition of f ( a ) and f + ( a ) ensure that f (5) = lim h → { 6 (5 + h ) } 1 h = lim h → µ h h ¶ = 1 , while f + (5) = lim h → + ‰ 1 6 (5 + h ) 1 ¾ h = lim h → + 1 (1 h ) h (1 h ) = lim h → + 1 1 h = 1 . Consequently, f (5) = 1 , f + (5) = 1 . 003 (part 2 of 2) 10 points (ii) Use your results from part (i) to deter mine if f (5) exists, and if it does, find its value. 1. f (5) = 0 2. f (5) does not exist correct 3. f (5) = 1 4. f (5) = 1 5. f (5) = 5 6. f (5) = 2 Mahon, Kevin – Homework 5 – Due: Sep 29 2005, 3:00 am – Inst: Edward Odell 3 7. f (5) = 2 Explanation: The derivative f (5) of f at x = 5 will exist if both the respective left and right hand derivatives f (5) and f + (5) exist and f (5) = f + (5). part (i) shows that these onesided derivatives exist, but are not equal. Consequently, f (5) does not exist . keywords: Stewart5e, left hand derivative, right hand derivative, derivative, piecewise defined function 004 (part 1 of 1) 10 points Find the value of f (4) when f ( x ) = x 3 / 2 4 x 1 / 2 ....
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 Fall '08
 schultz
 Derivative, Differential Calculus, Mahon, RLM, Edward Odell

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