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# EVNREV03 - Review Exercises for Chapter 3 50 Let f x Then f...

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Review Exercises for Chapter 3 Review Exercises for Chapter 3 437 50. Let Then tan 0.05 tan 0 sec 2 0 0.05 0 1 0.05 . f 0.05 f 0 f 0 dx f x tan x , x 0, dx 0.05, f x sec 2 x . 52. Propagated error relative error and the percent error dy y 100. dy y , f x x f x , 54. True, y x dy dx a 56. False Let Then and Thus, in this example. dy > y dy f x dx 1 2 1 3 3 2 . y f x x f x f 4 f 1 1 f x x , x 1, and x dx 3. 2. (a) (c) At least six critical numbers on 6, 6 . x 4 6 6 4 6 4 2 4 6 y f 4 f 4 3 (b) (d) Yes. Since and the Mean Value says that there exists at least one value c in such that (e) No, exists because f is continuous at (f) Yes, f is differentiable at x 2. 0, 0 . lim x 0 f x f c f 1 f 2 1 2 2 1 1 2 1. 2, 1 f 1 f 1 2, f 2 f 2 1 1 f 3 f 3 4 4 4. No critical numbers Left endpoint: Minimum Right endpoint: Maximum 2, 2 5 0, 0 1 x 2 1 3 2 f x x 1 2 x 2 1 3 2 2 x x 2 1 1 2 f x x x 2 1 , 0, 2 6. No. f is not differentiable at x 2. 8. No; the function is discontinuous at which is in the interval 2, 1 . x 0 10. c 2 f c 1 c 2 1 4 f b f a b a 1 4 1 4 1 3 4 3 1 4 f x 1 x 2 f x 1 x , 1 x 4 12. c 1 f c 1 2 c 2 3 2 f b f a b a 6 0 4 0 3 2 f x 1 2 x 2 f x x 2 x , 0 x 4

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438 Chapter 3 Applications of Differentiation 16. Critical number: x 1 g x 3 x 1 2 g x x 1 3 Interval Sign of Conclusion Increasing Increasing g x > 0 g x > 0 g x 1 < x < < x < 1 14. c 2 Midpoint of 0, 4 f c 4 c 3 5 f b f a b a 21 1 4 0 5 f x 4 x 3 f x 2 x 2 3 x 1 18. Critical numbers: x 4 , x 5 4 f x cos x sin x f x sin x cos x , 0 x 2 Interval Sign of Conclusion Increasing Decreasing Increasing f x > 0 f x < 0 f x > 0 f x 5 4 < x < 2 4 < x < 5 4 0 < x < 4 20. Relative maximum: Relative minimum: 3 2 , 3 2 1 2 , 3 2 0 when x 1 2 , 3 2 g x 3 2 2 cos x 2 1 g x 3 2 sin x 2 1 , 0, 4 Test Interval Sign of Conclusion Increasing Decreasing Increasing g x > 0 g x < 0 g x > 0 g x 3 2 < x < 4 1 2 < x < 3 2 0 < x < 1 2 22. (a) Therefore, When y A A A 2 B 2 B B A 2 B 2 A 2 B 2 . v y 0, cos k m t B A 2 B 2 . sin k m t A A 2 B 2 0 when sin k m t cos k m t A B tan k m t A B . y A k m cos k m t B k m sin k m t y A sin k m t B cos k m t (b) Period: Frequency: 1 2 k m 1 2 k m 2 k m
Review Exercises for Chapter 3 439 24. Point of inflection: 0, 16 f x 6 x 0 when x 0. f x 3 x 2 12 f x x 2 2 x 4 x 3 12 x 16 Test Interval Sign of Conclusion Concave downward Concave upward f x > 0 f x < 0 f x 0 < x < < x < 0 26. Domain: is a relative minimum. 3, 5 h 3 1 8 > 0 h t 1 t 1 3 2 h t 1 2 t 1 0 t 3 1, h t t 4 t 1 28. 1 1 1 2 3 4 5 6 7 2 3 4 5 6 7 y x 30. x 2 Qs r x 2 2 Qs r Qs x 2 r 2 dC dx Qs x 2 r 2 0 C Q x s x 2 r 32. (a) (b) (c) when This is a maximum by the First Derivative Test. (d) No, because the coefficient term is negative. t 3 t 3.7. S t 0 1 14 0 25 S 0.1222 t 3 1.3655 t 2 0.9052 t 4.8429 34. lim x 2 x 3 x 2 5 lim x 2 x 3 5 x 2 0 36. lim x 3 x x 2 4 lim x 3 1 4 x 2 3 38. Horizontal asymptote: y 5 lim x 5 x 2 x 2 2 lim x 5 1 2 x 2 5 g x 5 x 2 x 2 2 40. Horizontal asymptotes: y ± 3 lim x 3 1 2 x 2 3 lim x 3 x x 2 2 lim x 3 x x x 2 2 x 2 lim x 3 1 2 x 2 3 lim x 3 x x 2 2 lim x 3 x x x 2 2 x 2 f x 3 x x 2 2

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EVNREV03 - Review Exercises for Chapter 3 50 Let f x Then f...

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