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chapter_3-1 - Chapter 3 Additional Applications of the...

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91 Chapter 3 Additional Applications of the Derivative 3.1 Increasing and Decreasing Functions; Relative Extrema 2. The graph is rising for 3, x < so the first derivative is positive. It is falling if 3 x > which indicates that the derivative is negative. 4. The graph is falling for 1 x < and 35 , x << so the first derivative is negative. It is rising if 13 x and 5 x > which indicates that the derivative is positive. 6. The graph of the function is increasing for all x implying the derivative is positive for all x . Thus the correct match is graph C. 8. The derivative of this function is first negative (the graph of f ( x ) is increasing), then positive ( f ( x ) then decreases) and lastly negative ( f ( x ) again increases.) The derivative behavior matches graph A. 10. 32 2 () 3 1 () 3 6 3( 2) 0 when 2 and 0. ft t t ft t t t t tt =+ + =+ = + = =− = –5 –4 –3 –2 –1 1 2 3 4 5 –50 –40 –30 –20 –10 50 40 30 20 10 ( ) is increasing on 2 and 0 . ( ) is decreasing on 2 0. t t t <− < −<< 12. 3 2 9 2 3 9 ( 3 ) ( 3 ) 0 when 3 and 3. ( ) is increasing on 3 and 3 . ( ) is decreasing on 3 3. x fx x fx x x x xx f x x + + = = < −< < –10 –8 –6 –4 –2 2 4 6 8 10 –100 –80 –60 –40 –20 100 80 60 40 20 14. 53 5 x x 42 2 ( ) 15 15 15 ( 1)( 1) 0 x x x + = when x = 1, x = 0, and x = 1. –5 –4 –3 –2 –1 1 2 3 4 5 –20 –10 20 10 ( ) is increasing on 1 and 1 . ( ) is decreasing on 1 0 and 01 . x x x x <
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92 Chapter 3. Additional Applications of the Derivative 16. 22 2 2 23 11 () 1( 1 ) 2( 1 ) 0 (1 ) gt tt t =− ++ −− == + when 0, 1 t and 1. –10 –8 –6 –4 –2 246 8 10 0.25 0.5 g ( t ) is increasing on 1 t <− and 01 t << . g ( t ) is decreasing on 10 t −< < and 1 t < . 18. First note that 2 6 ( 3 ) ( 2 ) fx x x x x −= + so f ( x ) is only defined for 32 x −≤ ≤ . 2 12 0 26 x xx when 1 2 x . –3 –2 –1 1 2 0.25 0.5 f ( x ) is increasing on 1 3 2 x −< <− . f ( x ) is decreasing on 1 2 2 x −<< . 20. 2 3 (3 ) 3 0 ) t ft t t t = + + when 3 t = . Also note f ( t ) is not defined at 3 t . 8– 6 –4 –2 2 –4 –2 f ( t ) is increasing on 33 t . f ( t ) is decreasing on 3 t and 3 t < . 22. 2 2 3 1 2 () 2 Gx x x x x =+ The derivative is never 0 but is undefined at 0 x = . 6– 5 –4 –3 –2 –1 2 1 3 4 5 6 –10 –20 –30 –40 40 30 20 10 G ( x ) is increasing on 0 x < . G ( x ) is decreasing on 0 x < . 24. () 3 2 4 7 2 4 x x x =−+ 2 ( ) 324 144 12 12( 9)( 3) 0 x x + =−− = when 3 and 9. (3) 432 and (9) 0. f f = = The point (3, 432) is a relative maximum while (9, 0) is a relative minimum.
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Chapter 3. Additional Applications of the Derivative 93 –4 –2 8 10 12 246 14 16 –1,000 –500 1,000 500 26. 654 ( ) 10 24 15 3 ft t t t =+++ 54 3 22 ( ) 60 120 60 60 ( 1) 0 t t t tt =+ + = when 1 and 0. ( 1) 4 and (0) 3. f f =− = = = The function is rising/falling according to the figure below. If 0 ( ) 0 else ( ) 0 tf t f t ′′ >> which indicates that (0, 3) is a relative minimum and ( 1 4) is not a relative extremum. 2– 1 2 1 10 8 6 4 2 28. 3 2 () 3 ( 1 ) () 3 ( 1 ) 0 Fx x x =− + + = when 1. ( 1) 3. xF = The function is rising/falling according to the figure below. ( 1, 3) is neither a relative maximum nor a relative minimum.
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chapter_3-1 - Chapter 3 Additional Applications of the...

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