Quest HW 9 - roberts (eer474) Quest HW 9 seckin (56425) 1...

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Unformatted text preview: roberts (eer474) Quest HW 9 seckin (56425) 1 This print-out should have 14 questions. Multiple-choice 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.

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Quest HW 9 - roberts (eer474) Quest HW 9 seckin (56425) 1...

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