Calculus: Early Transcendentals, by Anton, 7th Edition,ch05

Calculus - Early Transcendentals

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150 CHAPTER 5 The Derivative in Graphing and Applications EXERCISE SET 5.1 1. (a) f 0 > 0 and f 0 > 0 y x (b) f 0 > 0 and f 0 < 0 y x (c) f 0 < 0 and f 0 > 0 y x (d) f 0 < 0 and f 0 < 0 y x 2. (a) y x (b) y x (c) y x (d) y x 3. A : dy/dx < 0 ,d 2 y/dx 2 > 0 B : dy/dx > 0 2 y/dx 2 < 0 C : dy/dx < 0 2 y/dx 2 < 0 4. A : dy/dx < 0 2 y/dx 2 < 0 B : dy/dx < 0 2 y/dx 2 > 0 C : dy/dx > 0 2 y/dx 2 < 0 5. An inflection point occurs when f 0 changes sign: at x = 1 , 0 , 1 and 2. 6. (a) f (0) <f (1) since f 0 > 0on(0 , 1). (b) f (1) >f (2) since f 0 < 0on(1 , 2). (c) f 0 (0) > 0 by inspection. (d) f 0 (1) = 0 by inspection. (e) f 0 (0) < 0 since f 0 is decreasing there. (f) f 0 (2) = 0 since f 0 has a minimum there. 7. (a) [4 , 6] (b) [1 , 4] and [6 , 7] (c) (1 , 2) and (3 , 5) (d) (2 , 3) and (5 , 7) (e) x =2 , 3 , 5
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Exercise Set 5.1 151 8. (1 , 2) (2 , 3) (3 , 4) (4 , 5) (5 , 6) (6 , 7) f 0 + + f 0 + + + 9. (a) f is increasing on [1 , 3] (b) f is decreasing on ( −∞ , 1] , [3 , + ] (c) f is concave up on ( −∞ , 2) , (4 , + ) (d) f is concave down on (2 , 4) (e) points of inflection at x =2 , 4 10. (a) f is increasing on ( −∞ , + ) (b) f is nowhere decreasing (c) f is concave up on ( −∞ , 1) , (3 , + ) (d) f is concave down on (1 , 3) (e) f has points of inflection at x =1 , 3 11. f 0 ( x )=2 x 5 (a) [5 / 2 , + ) (b) ( −∞ , 5 / 2] f 0 ( x (c) ( −∞ , + ) (d) none (e) none 12. f 0 ( x )= 2( x +3 / 2) (a) ( −∞ , 3 / 2] (b) [ 3 / 2 , + ) f 0 ( x 2 (c) none (d) ( −∞ , + ) (e) none 13. f 0 ( x )=3( x +2) 2 (a) ( −∞ , + ) (b) none f 0 ( x )=6( x (c) ( 2 , + ) (d) ( −∞ , 2) (e) 2 14. f 0 ( x ) = 3(4 x 2 ) (a) [ 2 , 2] (b) ( −∞ , 2], [2 , + ) f 0 ( x 6 x (c) ( −∞ , 0) (d) (0 , + ) (e) 0 15. f 0 ( x )=12 x 2 ( x 1) (a) [1 , + ) (b) ( −∞ , 1] f 0 ( x )=36 x ( x 2 / 3) (c) ( −∞ , 0), (2 / 3 , + ) (d) (0 , 2 / 3) (e) 0 , 2 / 3 16. f 0 ( x )=4 x ( x 2 4) (a) [ 2 , 0], [2 , + ) (b) ( −∞ , 2], [0 , 2] f 0 ( x ) = 12( x 2 4 / 3) (c) ( −∞ , 2 / 3), (2 / 3 , + ) (d) ( 2 / 3 , 2 / 3) (e) 2 / 3, 2 / 3 17. f 0 ( x 4 x ( x 2 2 f 0 ( x 4 3 x 2 2 ( x 2 3 (a) [0 , + ) (b) ( −∞ , 0] (c) ( p 2 / 3 , p 2 / 3) (d) ( −∞ , p 2 / 3), ( p 2 / 3 , + ) (e) p 2 / 3 , p 2 / 3 18. f 0 ( x 2 x 2 ( x 2 2 f 0 ( x 2 x ( x 2 6) ( x 2 3 (a) [ 2 , 2] (b) ( −∞ , 2], [ 2 , + ) (c) ( 6 , 0), ( 6 , + ) (d) ( −∞ , 6), (0 , 6) (e) 6 , 0 , 6 19. f 0 ( x 1 3 ( x 2 / 3 (a) ( −∞ , + ) (b) none f 0 ( x 2 9 ( x 5 / 3 (c) ( −∞ , 2) (d) ( 2 , + ) (e) 2
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152 Chapter 5 20. f 0 ( x )= 2 3 x 1 / 3 (a) [0 , + ) (b) ( −∞ , 0] f 0 ( x 2 9 x 4 / 3 (c) none (d) ( −∞ , 0), (0 , + ) (e) none 21. f 0 ( x 4( x +1) 3 x 2 / 3 (a) [ 1 , + ) (b) ( −∞ , 1] f 0 ( x 4( x 2) 9 x 5 / 3 (c) ( −∞ , 0), (2 , + ) (d) (0 , 2) (e) 0 , 2 22. f 0 ( x 4( x 1 / 4) 3 x 2 / 3 (a) [1 / 4 , + ) (b) ( −∞ , 1 / 4] f 0 ( x 4( x +1 / 2) 9 x 5 / 3 (c) ( −∞ , 1 / 2), (0 , + ) (d) ( 1 / 2 , 0) (e) 1 / 2 , 0 23. f 0 ( x xe x 2 / 2 (a) ( −∞ , 0] (b) [0 , + ) f 0 ( x )=( 1+ x 2 ) e x 2 / 2 (c) ( −∞ , 1), (1 , + ) (d) ( 1 , 1) (e) 1 , 1 24. f 0 ( x )=(2 x 2 e x 2 (a) ( −∞ , + ) (b) none f 0 ( x )=2 x (2 x 2 +3) e x 2 (c) (0 , + ) (d) ( −∞ , 0) (e) 0 25. f 0 ( x 2 x x 2 (a) [0 , + ) (b) ( −∞ , 0] f 0 ( x 1 x 2 (1 + x 2 ) 2 (c) ( 1 , 1) (d) ( −∞ , 1) , (1 , + ) (e) 1 , 1 26. f 0 ( x x (2 ln x (a) [ e 1 / 2 , + ) (b) (0 ,e 1 / 2 ] f 0 ( x )=2ln x +3 (c) ( e 3 / 2 , + ) (d) (0 3 / 2 ) (e) e 3 / 2 27. f 0 ( x sin x f 0 ( x cos x (a) [ π, 2 π ] (b) [0 ] (c) ( π/ 2 , 3 2) (d) (0 ,π/ 2), (3 2 , 2 π ) (e) 2, 3 2 1 -1 02 p 28. f 0 ( x ) = 2 sin 4 x f 0 ( x )=8cos4 x (a) (0 4], [ 2 , 3 4] (b) [ 4 2], [3 4 ] (c) (0 8), (3 8 , 5 8), (7 8 ) (d) ( 8 , 3 8), (5 8 , 7 8) (e) 8, 3 8, 5 8, 7 8 1 0 0 p
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Exercise Set 5.1 153 29. f 0 ( x ) = sec 2 x f 0 ( x )=2sec 2 x tan x (a) ( π/ 2 ,π/ 2) (b) none (c) (0 2) (d) ( 2 , 0) (e) 0 10 -10 ^6 30. f 0 ( x )=2 csc 2 x f 0 ( x )=2csc 2 x cot x =2 cos x sin 3 x (a) [ 4 , 3 4] (b) (0 4] , [3 4 ) (c) (0 2) (d) ( 2 ) (e) 2 8 -2 0 p 31. f 0 ( x
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Calculus: Early Transcendentals, by Anton, 7th Edition,ch05...

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