This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: vasquez (mpv244) HW09 Schultz (56445) 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 Determine if the function f ( x ) = x x + 12 satisfies the hypotheses of Rolles Theorem on the interval [ 12 , 0], and if it does, find all numbers c satisfying the conclusion of that theorem. 1. c = 8 correct 2. c = 9 3. c = 8 , 9 4. c = 5 5. c = 8 , 8 6. hypotheses not satisfied Explanation: Rolles Theorem says that if f is 1. continuous on [ a, b ] , 2. differentiable on ( a, b ) , and 3. f ( a ) = f ( b ) = 0, then there exists at least one c , a < c < b , such that f ( c ) = 0. Now the given function f ( x ) = x x + 12 , is defined for all x 12, is continuous on [ 12 , ), and differentiable on ( 12 , ). In addition f ( 12) = f (0) = 0 . In particular, therefore, Rolles theorem ap plies to f on [ 12 , 0]. On the other hand, by the Product and Chain Rules, f ( x ) = x + 12+ x 2 x + 12 = 3 x + 24 2 x + 12 . Thus there exists c, 12 < c < 0, such that f ( c ) = 3 c + 24 2 c + 12 = 0 , in which case c = 8 . 002 10.0 points Let f be a function defined on [0 , 1] such that f (0) = 1 , f (1) = 2 . Consider the following properties that f might have: A. f ( x ) =  3 x 1  ; B. f is cont. on [0 , 1] , ; C. f ( x ) = braceleftBigg x 2 + 1 , x negationslash = 1 / 2, 1 , x = 1 / 2 ; which properties ensure that f ( c ) = 1 for some c in (0 , 1)? 1. A and B only 2. B only 3. C only 4. A only 5. none of them correct 6. B and C only 7. all of them 8. A and C only Explanation: The Mean Value Theorem (MVT) says: vasquez (mpv244) HW09 Schultz (56445) 2 If f is a function continuous on [ a, b ] and differentiable on ( a, b ) , then f ( c ) = f ( b ) f ( a ) b a for at least one c in ( a, b ) . If one or more of the hypotheses fail, then there need not exist any such c . A. No c : f ( x ) = braceleftBigg 3 , x < 1 / 3, 3 , x > 1 / 3. B. No c : f ( x ) = braceleftBigg 1 , x < 1 / 2 2 x, 1 / 2 x 1. C. No c : ( f ( x ) = 2 x, x negationslash = 1 / 2). 003 10.0 points How many real roots does the equation x 5 + 5 x + 3 = 0 have? 1. exactly four real roots 2. exactly two real roots 3. exactly three real roots 4. exactly one real root correct 5. no real roots Explanation: Define f by f ( x ) = x 5 + 5 x + 3 . Then the roots of the equation x 5 + 5 x + 3 = 0 are the xintercepts of the graph of f . Now f ( x ) x 5 for  x  large, so f ( x ) as x , while f ( x ) as x ....
View
Full
Document
 Spring '09
 RUTH
 Chemistry

Click to edit the document details