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Unformatted text preview: Granillo, Yvette 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 , 4) [ ( 4 , 3) [ (3 , 10) 2. ( 10 , 4) [ ( 4 , 10) 3. ( 10 , 3) [ (3 , 10) correct 4. ( 10 , 4] [ [3 , 10) 5. ( 10 , 10) 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 , 3 in the case of g . But the sum of continuous functions is again continuous, so f + g is certainly continuous on ( 10 , 4) [ ( 4 , 3) [ (3 , 10) . The only question is what happens at x = 4 , 3. 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 ) } = 2 = f ( 4) + g ( 4) = lim x  4+ { f ( x ) + g ( x ) } , while at x = 3, lim x 3 { f ( x ) + g ( x ) } = 7 6 = 5 = lim x 3+ { f ( x ) + g ( x ) } . Thus, f + g is continuous at x = 4, but not at x = 3. Consequently, on ( 10 , 10) the sum f + g is continuous at all x in ( 10 , 3) [ (3 , 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 Granillo, Yvette 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. x = 3 , 3 correct 3. none of these 4. x = 3 5. no values of x 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 = 3. But at x = 3 f ( 3) = 9 6 = lim x  3 f ( x ) = 6 , while at x = 3 lim x 3 f ( x ) = 6 6 = lim x 3+ f ( x ) = 4 . Consequently, on ( 6 , 6) the function f fails to be continuous only at at x = 3 , 3 . keywords: Stewart5e, 003 (part 1 of 1) 10 points If f and g are continuous functions such that lim x 3 [7 f ( x ) g ( x )] = 9 , f (3) = 2 , find the value of g (3). 1. g (3) = 14 2. g (3) = 5 correct 3. g (3) = 9 4. g (3) = 2 5. g (3) = 23 Explanation: Since f and g are continuous functions, lim x 3 (7 f ( x ) g ( x )) = 7 lim x 3 f ( x ) lim x 3 g ( x ) = 7 f (3) g (3) = 14 g (3) ....
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This test prep was uploaded on 04/16/2008 for the course M 408k taught by Professor Schultz during the Spring '08 term at University of Texas at Austin.
 Spring '08
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