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MAC 2311 Test Three Review Answers, Fall 2010 1. f ( x ): critical numbers: x = ± 1, x = 0, x = ± p 3 relative minimum: f ( ² 1) = ² 3 p 2, relative maximum: f (1) = 3 p 2 g ( x ): critical numbers: x = ² 2, x = 4 only relative maximum: f ( ² 2) = ² 4, relative minimum: f (4) = 8 2. b) 1) c = 3 ± 2 2) not possible; f is not di±erentiable at x = 0 3) not possible; f is not continuous at x = ± 2 c) Use the Intermediate Value Theorem to show that the equation 3 x ² cos x ² 1 has at least one real root; assume there are two roots a and b and use Rolle’s Theorem to get a contradiction (see class notes, homework for similar example). d) c = 5 ² p 5 e) c = e ² 1 f) Show that f 0 ( x ) = 0, so that f ( x ) = C where C is a constant. Then evaluate f at a point to ²nd C . 3. Use the Mean Value Theorem on [0 ; 6] to ²nd f (6) ³ ² 8. 4. maximum: f (1) = 1, minimum: f (2) = 4 ² 8 ln 2 5. a) 1 b) 2 c) ²1 d) 1 2 e) e 2 f) 0 6. on [2 ; 4]: maximum is f (2) = 8 and minimum is f (3) = 6 : 75 on (1 ; 1 ): minimum is f (3) = 6 : 75 and there is no maximum value 7. L ( x ) = 1 e x 8. L ( x ) = 2 + 1 4 ( x ² 2) = 1 4 x + 3 2 ; 3 p 8 : 228 ´ 2 : 019 9. a) 0.3225 b. 0.3 10. maximum error: (80 + 10

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