Homework #4 - jiwani(amj566 – Homework04 – Fouli...

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Unformatted text preview: jiwani (amj566) – Homework04 – Fouli – (58395) 1 This print-out should have 27 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 product, fg , of f and g is continuous, expressing your answer in interval notation. 1. ( − 10 , − 2) uniondisplay ( − 2 , 4) uniondisplay (4 , 10) 2. ( − 10 , 10) correct 3. ( − 10 , 4) uniondisplay (4 , 10) 4. ( − 10 , − 2] uniondisplay [4 , 10) 5. ( − 10 , − 2) uniondisplay ( − 2 , 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 , − 2 in the case of g . But the product of continuous functions is again continuous, so fg is certainly continuous on ( − 10 , − 2) uniondisplay ( − 2 , 4) uniondisplay (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 ) } = − 18 = f (4) g (4) = lim x → 4+ { f ( x ) g ( x ) } , while at x = − 2, lim x →− 2 − { f ( x ) g ( x ) } = 0 = f ( − 2) g ( − 2) = lim x →− 2+ { f ( x ) g ( x ) } . Thus, fg is continuous at x = 4 and at x = − 2. Consequently, the product fg is continuous at all x in ( − 10 , 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) = − 8 2. f (4) = 8 correct 3. f (4) = 16 4. f (4) = 4 5. f (4) = − 16 6. f (4) = − 4 Explanation: jiwani (amj566) – Homework04 – Fouli – (58395) 2 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 . 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 = − 6 , − 1 2. x = − 6 , − 1 , 4 correct 3. x = − 6 , 4 4. f is continuous everywhere 5. x = − 1 , 4 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....
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Homework #4 - jiwani(amj566 – Homework04 – Fouli...

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