M408K Hm. 4 Solutions

# M408K Hm. 4 Solutions - Mahon, Kevin Homework 4 Due: Sep 22...

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Unformatted text preview: Mahon, Kevin Homework 4 Due: Sep 22 2005, 3:00 am Inst: Edward Odell 1 This print-out should have 24 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 Functions f and g are defined on (- 10 , 10) by their respective graphs in 2 4 6 8- 2- 4- 6- 8 4 8- 4- 8 f g Find all values of x where the sum, f + g , of f and g is continuous, expressing your answer in interval notation. 1. (- 10 ,- 2) [ (- 2 , 4) [ (4 , 10) 2. (- 10 , 4) [ (4 , 10) 3. (- 10 ,- 2] [ [4 , 10) 4. (- 10 , 10) 5. (- 10 ,- 2) [ (- 2 , 10) correct Explanation: Since f and g are piecewise linear, they are continuous individually on (- 10 , 10) except at their jumps, i.e. , at x = 4 in the case of f and x = 4 ,- 2 in the case of g . But the sum of continuous functions is again continuous, so f + g is certainly continuous on (- 10 ,- 2) [ (- 2 , 4) [ (4 , 10) . The only question is what happens at x = 4 ,- 2. To do that we have to check if lim x x- { f ( x ) + g ( x ) } = f ( x ) + g ( x ) = lim x x + { f ( x ) + g ( x ) } . Now at x = 4, lim x 4- { f ( x ) + g ( x ) } = 1 = f (4) + g (4) = lim x 4+ { f ( x ) + g ( x ) } , while at x =- 2, lim x - 2- { f ( x ) + g ( x ) } =- 2 6 =- 5 = lim x - 2+ { f ( x ) + g ( x ) } . Thus, f + g is continuous at x = 4, but not at x =- 2. Consequently, on (- 10 , 10) the sum f + g is continuous at all x in (- 10 ,- 2) [ (- 2 , 10) . keywords: Stewart5e, 002 (part 1 of 1) 10 points Below is the graph of a function f . 2 4 6- 2- 4- 6 2 4 6 8- 2- 4 Mahon, Kevin Homework 4 Due: Sep 22 2005, 3:00 am Inst: Edward Odell 2 Use the graph to determine all the values of x on (- 6 , 6) at which f fails to be continuous. 1. x =- 3 , 2 correct 2. no values of x 3. x =- 3 4. x = 2 5. none of these Explanation: Since f ( x ) is defined for all values of x on (- 6 , 6), the only values of x in (- 6 , 6) at which the function f is discontinuous are those for which lim x x f ( x ) 6 = f ( x ) or lim x x- f ( x ) 6 = lim x x + f ( x ) . The only possible candidates here are x =- 3 and x = 2. But at x =- 3 f (- 3) = 9 6 = lim x - 3 f ( x ) = 2 , while at x = 2 lim x 2- f ( x ) = 6 6 = lim x 2+ f ( x ) = 0 . Consequently, on (- 6 , 6) the function f fails to be continuous only at at x =- 3 , 2 . keywords: Stewart5e, 003 (part 1 of 1) 10 points If f and g are continuous functions such that lim x 5 [5 f ( x )- g ( x )] = 3 , f (5) = 2 , find the value of g (5). 1. g (5) = 2 2. g (5) = 10 3. g (5) = 13 4. g (5) = 7 correct 5. g (5) = 3 Explanation: Since f and g are continuous functions, lim x 5 (5 f ( x )- g ( x )) = 5 lim x 5 f ( x )- lim x 5 g ( x ) = 5 f (5)- g (5) = 10- g (5) ....
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## M408K Hm. 4 Solutions - Mahon, Kevin Homework 4 Due: Sep 22...

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