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Unformatted text preview: syed (sms3768) 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 [ 3 , 6], and if it can, find all numbers c satisfying the conclusion of that theorem. 1. c = 3 , 3 2 2. c = 3 3. Rolles Theorem not applicable correct 4. c = 1 5. c = 3 , 15 6. c = 3 2 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 [ 3 , 6]. 002 10.0 points Determine if the function f ( x ) = x x + 15 satisfies the hypotheses of Rolles Theorem on the interval [ 15 , 0], and if it does, find all numbers c satisfying the conclusion of that theorem. 1. c = 6 2. c = 10 correct 3. hypotheses not satisfied 4. c = 10 , 10 5. c = 10 , 11 6. c = 11 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 + 15 , is defined for all x 15, is continuous on [ 15 , ), and differentiable on ( 15 , ). In addition f ( 15) = f (0) = 0 . In particular, therefore, Rolles theorem ap- plies to f on [ 15 , 0]. On the other hand, by the Product and Chain Rules, f ( x ) = x + 15+ x 2 x + 15 = 3 x + 30 2 x + 15 . syed (sms3768) Quest HW 9 seckin (56425) 2 Thus there exists c, 15 < c < 0, such that f ( c ) = 3 c + 30 2 c + 15 = 0 , in which case c = 10 . 003 10.0 points Determine if the function f ( x ) = x 3 x + 2 satisfies the hypotheses of the Mean value Theorem (MVT) on the interval [ 1 , 2]. If it does, find all possible values of c satis- fying the conclusion of the MVT....
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This note was uploaded on 04/29/2010 for the course MATH 408K taught by Professor Seckin during the Spring '10 term at University of Texas-Tyler.
- Spring '10