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Unformatted text preview: Mahon, Kevin Homework 4 Due: Sep 22 2005, 3:00 am Inst: Edward Odell 1 This printout should have 24 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 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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 Fall '08
 schultz
 Differential Calculus

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