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Unformatted text preview: Version 018 – M408K Final Exam – Zheng – (57255) 1 This printout should have 25 questions. Multiplechoice 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 ) , Version 018 – M408K Final Exam – Zheng – (57255) 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 Let F be the function defined by F ( x ) = x 2 − 81  x − 9  . Determine if lim x → 9 F ( x ) exists, and if it does, find its value. 1. limit = − 18 correct 2. limit does not exist 3. limit = − 9 4. limit = 18 5. limit = 9 Explanation: After factorization, x 2 − 81  x − 9  = ( x + 9)( x − 9)  x − 9  . But, for x < 9,  x − 9  = − ( x − 9) . Thus F ( x ) = − ( x + 9) , x < 9 , By properties of limits, therefore, the limit exists and lim x → 9 F ( x ) = − 18 . 003 10.0 points Below is the graph of a function f . 2 4 6 − 2 − 4 − 6 2 4 6 8 − 2 − 4 Use the graph to determine lim x → − 5 f ( x ). 1. lim x → − 5 f ( x ) = 2 2. lim x → − 5 f ( x ) = 6 correct 3. lim x → − 5 f ( x ) does not exist 4. lim x → − 5 f ( x ) = 4 5. lim x → − 5 f ( x ) = 8 Explanation: From the graph it is clear the f has both a left hand limit and a right hand limit at x = − 5; in addition, these limits coincide. Thus lim x →− 5 f ( x ) = 6. Version 018 – M408K Final Exam – Zheng – (57255) 3 004 10.0 points For which of the following functions f and corresponding numbers a is the limit lim h → (1 + h ) 5 − 1 h the value of f ′ ( a )? 1. f ( x ) = ( x + 1) 5 , a = 1 2. f ( x ) = x 5 , a = 0 3. f ( x ) = x 5 , a = 5 4. f ( x ) = ( x + 1) 5 , a = 5 5. f ( x ) = ( x − 1) 5 , a = 1 6. f ( x ) = x 5 , a = 1 correct Explanation: By definition f ′ ( a ) = lim h → f ( a + h ) − f ( a ) h . When f ( a + h ) − f ( a ) h = (1 + h ) 5 − 1 h , therefore, inspection shows that f ( x ) = x 5 , a = 1 . 005 10.0 points Determine if lim x → e 4 x − 1 sin6 x exists, and if it does, find its value....
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 Fall '08
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