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Quest HW 3-solutions - syed(sms3768 Quest HW 3 seckin(56425...

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syed (sms3768) – Quest HW 3 – seckin – (56425) 1 This print-out should have 21 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 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) uniondisplay (2 , 10) correct 2. ( 10 , 5] uniondisplay [2 , 10) 3. ( 10 , 5) uniondisplay ( 5 , 10) 4. ( 10 , 5) uniondisplay ( 5 , 2) uniondisplay (2 , 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 = 5 in the case of f and x = 5 , 2 in the case of g . But the sum of continuous functions is again continuous, so f + g is certainly continuous on ( 10 , 5) uniondisplay ( 5 , 2) uniondisplay (2 , 10) . The only question is what happens at x 0 = 5 , 2. To do that we have to check if lim x x 0 { f ( x ) + g ( x ) } = f ( x 0 ) + g ( x 0 ) = lim x x 0 + { f ( x ) + g ( x ) } . Now at x 0 = 5, lim x → − 5 { f ( x ) + g ( x ) } = 4 = f ( 5) + g ( 5) = lim x → − 5+ { f ( x ) + g ( x ) } , while at x 0 = 2, lim x 2 { f ( x ) + g ( x ) } = 8 negationslash = 6 = lim x 2+ { f ( x ) + g ( x ) } . Thus, f + g is continuous at x = 5, but not at x = 2. Consequently, on ( 10 , 10) the sum f + g is continuous at all x in ( 10 , 2) uniondisplay (2 , 10) . 002 10.0 points If the function f is continuous everywhere and f ( x ) = x 2 16 x 4 when x negationslash = 4, find the value of f (4). 1. f (4) = 4 2. f (4) = 4 3. f (4) = 8 4. f (4) = 8 correct 5. f (4) = 16 6. f (4) = 16
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syed (sms3768) – Quest HW 3 – seckin – (56425) 2 Explanation: Since f is continuous at x = 4, f (4) = lim x 4 f ( x ) . But, after factorization, x 2 16 x 4 = ( x 4)( x + 4) x 4 = x + 4 , whenever x negationslash = 4. Thus f ( x ) = x + 4 for all x negationslash = 4. Consequently, f (4) = lim x 4 ( x + 4) = 8 . 003 10.0 points Below is the graph of a function f . -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 2 4 6 2 4 6 2 4 2 4 Use this graph to determine all the values of x at which f fails to be continuous on ( 8 , 8). 1. x = 5 , 1 2. x = 5 , 3 3. x = 1 , 3 4. x = 5 , 1 , 3 correct 5. f is continuous everywhere Explanation: The function f is continuous at a point a in ( 8 , 8) when (i) f ( a ) is defined, (ii) lim x a f ( x ) exists, and (iii) lim x a f ( x ) = f ( a ). We check where one or more of these condi- tions fails. (i) This fails at a = 3. The only other possible candidates are a = 5 and x 0 = 1: (ii) At x 0 = 5 lim x →− 5 f ( x ) = 5 negationslash = lim x →− 5+ f ( x ) = 2 so the limit does not exist; while (iii) at x 0 = 1, f ( 1) = 1 negationslash = lim x → − 1 f ( x ) = 2 , so the limit exists but does not have value f ( a ). Consequently, f fails to be continuous only at x = 5 , 1 , 3 on ( 8 , 8). 004 10.0 points Determine which of the following could be the graph of f near the origin when f ( x ) = x 2 7 x + 10 2 x , x negationslash = 2 , 2 , x = 2 .
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