calculus homework 4

# calculus homework 4 - obiahu (vo879) HW04 Schultz (56395) 1...

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Unformatted text preview: obiahu (vo879) HW04 Schultz (56395) 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 Determine which of the following could be the graph of f near the origin when f ( x ) = x 2 x 2 x 2 , x negationslash = 2 , 4 , x = 2 . 1. 2. 3. 4. 5. 6. correct Explanation: Since x 2 x 2 x 2 = ( x 2)( x + 1) x 2 = x + 1 , for x negationslash = 2, we see that f is linear on ( , 2) uniondisplay (2 , ) , while lim x 2 f ( x ) = 3 negationslash = f (2) . Thus the graph of f will be a straight line of slope 1, having a hole at x = 2. This eliminates four of the possible graphs. But the two remaining graphs are the same except that in one f (2) > lim x 2 f ( x ) , obiahu (vo879) HW04 Schultz (56395) 2 while in the other f (2) < lim x 2 f ( x ) . Consequently, must be the graph of f near the origin. 002 10.0 points Find all values of x at which the function f defined by f ( x ) = x 2 x 2 7 x + 12 is continuous, expressing your answer in in- terval notation. 1. ( , 4) ( 4 , 3) ( 3 , ) 2. ( , 3) (3 , ) 3. ( , 3) (3 , 4) (4 , ) correct 4. ( , 4) (4 , ) 5. ( , 3) ( 3 , 4) (4 , ) Explanation: After factorization the denominator be- comes x 2 7 x + 12 = ( x 3)( x 4) , so f can be written as f ( x ) = x 2 ( x 3)( x 4) . Being a rational function, it will be contin- uous everywhere except at the zeros of the denominator since it will not be defined at such points. Thus f is continuous everywhere except at x = 3 and x = 4. Hence it will continuous on ( , 3) (3 , 4) (4 , ) . 003 10.0 points Determine which (if any) of the following functions is not continuous at x = 7. 1. f ( x ) = 1 | x 5 | x 7 1 2 x < 7 2. f ( x ) = 1 x 5 x 7 1 2 x < 7 3. all continuous at x = 7 4. f ( x ) = braceleftBigg 21 2 x 7 x negationslash = 7 3 x = 7 5. f ( x ) = braceleftbigg | x 7 | x negationslash = 7 x = 7 6. f ( x ) = braceleftBigg 1 x 7 x negationslash = 7 7 x = 7 correct Explanation: A function f will be continuous at x = 7 when f (7) exists and lim x 7 f ( x ) = f (7) . Now f (7) exists for all the functions defined above; in addition, inspection shows that all these functions have the property lim x 7 f ( x ) = f (7) except for f ( x ) = braceleftBigg 1 x 7 x negationslash = 7 7 x = 7 . . obiahu (vo879) HW04 Schultz (56395) 3 Consequently, this function is the only one that is not continuous at x = 7....
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## This note was uploaded on 09/06/2010 for the course M 32795 taught by Professor Schultz during the Spring '10 term at University of Texas at Austin.

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calculus homework 4 - obiahu (vo879) HW04 Schultz (56395) 1...

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