M408K Hm. 5 Solutions

M408K Hm. 5 Solutions - Mahon, Kevin Homework 5 Due: Sep 29...

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Unformatted text preview: Mahon, Kevin Homework 5 Due: Sep 29 2005, 3:00 am Inst: Edward Odell 1 This print-out should have 22 questions. Multiple-choice 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 left-hand derivative f- ( a ) = lim h - f ( a + h )- f ( a ) h and right-hand 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 - - h h =- 1 , while f + (5) = lim h + 1 6- (5 + h )- 1 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 one-sided 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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M408K Hm. 5 Solutions - Mahon, Kevin Homework 5 Due: Sep 29...

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