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chapter_05 - 11:44 L24-CH05 Sheet number 1 Page number 153...

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January 27, 2005 11:44 L24-CH05 Sheet number 1 Page number 153 black 153 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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January 27, 2005 11:44 L24-CH05 Sheet number 2 Page number 154 black 154 Chapter 5 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 3 / 2) f ( x ) = 2 12. f ( x ) = 2(2 + x ) f ( x ) = 2 13. f ( x ) = 6(2 x + 1) 2 f ( x ) = 24(2 x + 1) 14. f ( x ) = 3(4 x 2 ) f ( x ) = 6 x (a) [ 2 , 2] (b) ( −∞ , 2], [2 , + ) (c) ( −∞ , 0) (d) (0 , + ) (e) 0 15. f ( x ) = 12 x 2 ( x 1) f ( x ) = 36 x ( x 2 / 3) 16. f ( x ) = x (4 x 2 15 x + 18) f ( x ) = 6( x 1)(2 x 3) 17. f ( x ) = 3( x 2 3 x + 1) ( x 2 x + 1) 3 f ( x ) = 6 x (2 x 2 8 x + 5) ( x 2 x + 1) 4 18. f ( x ) = x 2 2 ( x + 2) 2 f ( x ) = 2 x ( x 2 6) ( x + 2) 3 (a) ( −∞ , 2) , ( 2 , + ) (b) ( 2 , 2) (c) ( −∞ , 6), (0 , 6) (d) ( 6 , 0), ( 6 , + ) (e) none 19. f ( x ) = 2 x + 1 3( x 2 + x + 1) 2 / 3 f ( x ) = 2( x + 2)( x 1) 9( x 2 + x + 1) 5 / 3
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January 27, 2005 11:44 L24-CH05 Sheet number 3 Page number 155 black Exercise Set 5.1 155 20. f ( x ) = 4( x 1 / 4) 3 x 2 / 3 f ( x ) = 4( x + 1 / 2) 9 x 5 / 3
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