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Unformatted text preview: roberts (eer474) Quest HW 9 seckin (56425) 1 This printout should have 14 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine if Rolles Theorem can be ap plied to f ( x ) = x 2 + 3 x 18 x + 3 on the interval [ 6 , 3], and if it can, find all numbers c satisfying the conclusion of that theorem. 1. c = 3 , 3 2 2. c = 3 2 3. c = 3 4. c = 1 5. c = 3 , 15 6. Rolles Theorem not applicable correct Explanation: Rolles Theorem can be applied to the func tion F ( x ) = ( x a )( x b ) x m on the interval [ a, b ] so long as m does not belong to [ a, b ] because F is continuous and differentiable on ( , m ) uniondisplay ( m, ) . For the given function f we see that f ( x ) = ( x + 6)( x 3) x + 3 , so Rolles Theorem does not apply to f on the interval [ 6 , 3]. 002 10.0 points Determine if the function f ( x ) = x x + 21 satisfies the hypotheses of Rolles Theorem on the interval [ 21 , 0], and if it does, find all numbers c satisfying the conclusion of that theorem. 1. c = 14 correct 2. c = 14 , 15 3. c = 14 , 14 4. hypotheses not satisfied 5. c = 15 6. c = 8 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 + 21 , is defined for all x 21, is continuous on [ 21 , ), and differentiable on ( 21 , ). In addition f ( 21) = f (0) = 0 . In particular, therefore, Rolles theorem ap plies to f on [ 21 , 0]. On the other hand, by the Product and Chain Rules, f ( x ) = x + 21+ x 2 x + 21 = 3 x + 42 2 x + 21 . roberts (eer474) Quest HW 9 seckin (56425) 2 Thus there exists c, 21 < c < 0, such that f ( c ) = 3 c + 42 2 c + 21 = 0 , in which case c = 14 . 003 10.0 points Determine if the function f ( x ) = x + 1 x 2 satisfies the hypotheses of the Mean value Theorem (MVT) on the interval [1 , 4]....
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This note was uploaded on 11/08/2010 for the course MATH 408K taught by Professor Gualdani during the Spring '09 term at University of Texas at Austin.
 Spring '09
 Gualdani
 Differential Calculus

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