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Quest HW 9-solutions - syed(sms3768 – Quest HW 9 –...

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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 Rolle’s 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. Rolle’s Theorem not applicable correct 4. c = 1 5. c = 3 , 15 6. c = 3 2 Explanation: Rolle’s 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 Rolle’s 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 Rolle’s 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: Rolle’s 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, Rolle’s 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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Quest HW 9-solutions - syed(sms3768 – Quest HW 9 –...

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