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 and f > 0 y x (b) f > 0 and f < 0 y x (c) f < 0 and f > 0 y x (d) f < 0 and f < 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 , d 2 y/dx 2 < 0 C : dy/dx < 0 , d 2 y/dx 2 < 0 4. A : dy/dx < 0 , d 2 y/dx 2 < 0 B : dy/dx < 0 , d 2 y/dx 2 > 0 C : dy/dx > 0 , d 2 y/dx 2 < 0 5. An inflection point occurs when f changes sign: at x = 1 , 0 , 1 and 2. 6. (a) f (0) < f (1) since f > 0 on (0 , 1). (b) f (1) > f (2) since f < 0 on (1 , 2). (c) f (0) > 0 by inspection. (d) f (1) = 0 by inspection. (e) f (0) < 0 since f is decreasing there. (f) f (2) = 0 since f 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 + + f + + + 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 ( x ) = 2 x 5 f ( x ) = 2 12. f ( x ) = 2( x + 3 / 2) f ( x ) = 2 13. f ( x ) = 3( x + 2) 2 f ( x ) = 6( x + 2) 14. f ( x ) = 3(4 x 2 ) (a) [ 2 , 2] (b) ( −∞ , 2], [2 , + ) f ( x ) = 6 x (c) ( −∞ , 0) (d) (0 , + ) (e) 0 15. f ( x ) = 12 x 2 ( x 1) f ( x ) = 36 x ( x 2 / 3) 16. f ( x ) = 4 x ( x 2 4) f ( x ) = 12( x 2 4 / 3) 17. f ( x ) = 4 x ( x 2 + 2) 2 f ( x ) = 4 3 x 2 2 ( x 2 + 2) 3 18. f ( x ) = 2 x 2 ( x 2 + 2) 2 f ( x ) = 2 x ( x 2 6) ( x 2 + 2) 3 (a) [ 2 , 2] (b) ( −∞ , 2], [ 2 , + ) (c) ( 6 , 0), ( 6 , + ) (d) ( −∞ , 6), (0 , 6) (e) 6 , 0 , 6 19. f ( x ) = 1 3 ( x + 2) 2 / 3 (a) ( −∞ , + ) (b) none f ( x ) = 2 9 ( x + 2) 5 / 3 (c) ( −∞ , 2) (d) ( 2 , + ) (e) 2
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152 Chapter 5 20. f ( x ) = 2 3 x 1 / 3 f ( x ) = 2 9 x 4 / 3 21. f ( x ) = 4( x + 1) 3 x 2 / 3 f ( x ) = 4( x 2) 9 x 5 / 3 22. f ( x ) = 4( x 1 / 4) 3 x 2 / 3 f ( x ) = 4( x + 1 / 2) 9 x 5 / 3 23. f ( x ) = xe x 2 / 2 (a) ( −∞ , 0] (b) [0 , + ) f ( x ) = ( 1 + x 2 ) e x 2 / 2 (c) ( −∞ , 1), (1 , + ) (d) ( 1 , 1) (e) 1 , 1
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  • Critical Point, Fermat's theorem, relative minimum

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