This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View Full
Document
 Fall '08
 schultz

Click to edit the document details