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# Chapter_05 - 11:44 L24-CH05 Sheet number 1 Page number 153 black CHAPTER 5 The Derivative in Graphing and Applications EXERCISE SET 5.1 1(a f > 0

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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 > 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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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 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 3 / 2) f 0 ( x )=2 (a) [3 / 2 , + ) (b) ( −∞ , 3 / 2] (c) ( −∞ , + ) (d) nowhere (e) none 12. f 0 ( x )= 2(2 + x ) f 0 ( x 2 (a) ( −∞ , 2] (b) [ 2 , + ) (c) nowhere (d) ( −∞ , + ) (e) none 13. f 0 ( x ) = 6(2 x +1) 2 f 0 ( x ) = 24(2 x (a) ( −∞ , + ) (b) nowhere (c) ( 1 / 2 , + ) (d) ( −∞ , 1 / 2) (e) 1 / 2 14. f 0 ( x ) = 3(4 x 2 ) f 0 ( x 6 x (a) [ 2 , 2] (b) ( −∞ , 2], [2 , + ) (c) ( −∞ , 0) (d) (0 , + ) (e) 0 15. f 0 ( x )=12 x 2 ( x 1) f 0 ( x )=36 x ( x 2 / 3) (a) [1 , + ) (b) ( −∞ , 1] (c) ( −∞ , 0), (2 / 3 , + ) (d) (0 , 2 / 3) (e) 0 , 2 / 3 16. f 0 ( x x (4 x 2 15 x + 18) f 0 ( x )=6( x 1)(2 x 3) (a) [0 , + ), (b) ( −∞ , 0] (c) ( −∞ , 1), (3 / 2 , + ) (d) (1 , 3 / 2) (e) 1 , 3 / 2 17. f 0 ( x 3( x 2 3 x ( x 2 x 3 f 0 ( x 6 x (2 x 2 8 x +5) ( x 2 x 4 (a) [ 3 5 2 , 3+ 5 2 ] (b) ( −∞ , 3 5 2 ], [ 3+ 5 2 , + ) (c) (0 , 2 6 2 ), (2 + 6 2 , + ) (d) ( −∞ , 0), (2 6 2 , 2+ 6 2 ) (e) 0 , 2 6 / 2 , 6 / 2 18. f 0 ( x x 2 2 ( x +2) 2 f 0 ( x 2 x ( x 2 6) ( x 3 (a) ( −∞ , 2) , ( 2 , + ) (b) ( 2 , 2) (c) ( −∞ , 6), (0 , 6) (d) ( 6 , 0), ( 6 , + ) (e) none 19. f 0 ( x 2 x +1 3( x 2 + x 2 / 3 f 0 ( x 2( x + 2)( x 1) 9( x 2 + x 5 / 3 (a) [ 1 / 2 , + ) (b) ( −∞ , 1 / 2] (c) ( 2 , 1) (d) ( −∞ , 2) , (1 , + ) (e) 2 , 1
January 27, 2005 11:44 L24-CH05 Sheet number 3 Page number 155 black Exercise Set 5.1 155 20. f 0 ( x )= 4( x 1 / 4) 3 x 2 / 3 f 0 ( x 4( x +1 / 2) 9 x 5 / 3 (a) [1 / 4 , + ) (b) ( −∞ , 1 / 4] (c) ( −∞ , 1 / 2), (0 , + ) (d) ( 1 / 2 , 0) (e) 1 / 2 , 0 21. f 0 ( x 4( x 2 / 3 1) 3 x 1 / 3 f 0 ( x 4( x 5 / 3 + x ) 9 x 7 / 3 (a) [ 1 , 0] , [1 , + ) (b) ( −∞ , 1] , [0 , 1] (c) ( −∞ , 0), (0 , + ) (d) nowhere (e) none 22. f 0 ( x 2 3 x 1 / 3 1 f 0 ( x 2 9 x 4 / 3 (a) [ 1 , 0] , [1 , + ) (b) ( −∞ , 1] , [0 , 1] (c) ( −∞ , 0) , (0 , + ) (d) nowhere (e) none 23. f 0 ( x xe x 2 / 2 f 0 ( x )=( 1+ x 2 ) e x 2 / 2 (a) ( −∞ , 0] (b) [0 , + ) (c) ( −∞ , 1), (1 , + ) (d) ( 1 , 1) (e) 1 , 1 24. f 0 ( x )=(2 x 2 +1) e x 2 f 0 ( x )=2 x (2 x 2 +3) e x 2 (a) ( −∞ , + ) (b) none (c) (0 , + ) (d) ( −∞ , 0) (e) 0 25. f 0 ( x x x 2 +4 f 0 ( x x 2 4 ( x 2 +4) 2 (a) [0 , + ) (b) ( −∞ , 0] (c) ( 2 , +2) (d) ( −∞ , 2) , (2 , + ) (e) 2 , +2 26. f 0 ( x x 2 (1+3ln x ) f 0 ( x x (5+6ln x ) (a) [ e 1 / 3 , + ) (b) (0 ,e 1 / 3 ] (c) ( e 5 / 6 , + ) (d) (0 5 / 6 ) (e) e 5 / 6 27. f 0 ( x 2 x 1+( x 2 1) 2 f 0 ( x 2 3 x 4 2 x 2 2 [1+( x 2 1) 2 ] 2 (a)

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## This note was uploaded on 06/10/2010 for the course MATH 200-177 taught by Professor Richardwhite during the Spring '10 term at Drexel.

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Chapter_05 - 11:44 L24-CH05 Sheet number 1 Page number 153 black CHAPTER 5 The Derivative in Graphing and Applications EXERCISE SET 5.1 1(a f > 0

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