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Unformatted text preview: syed (sms3768) – Quest HW 3 – seckin – (56425) 1 This printout should have 21 questions. Multiplechoice 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 = − 5 , 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 = − 5, lim x →− 5 − { f ( x ) + g ( x ) } = − 4 = f ( − 5) + g ( − 5) = lim x →− 5+ { f ( x ) + g ( x ) } , while at x = 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 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 . 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 )....
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This note was uploaded on 04/29/2010 for the course MATH 408K taught by Professor Seckin during the Spring '10 term at University of TexasTyler.
 Spring '10
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