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# ch04 - Chapter 4 Derivatives Review of Prerequisite Skills...

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Review of Prerequisite Skills 1. f. i. 6. d. f. 9. c. d. 3 2 4 3 3 2 4 3 3 2 4 3 3 2 4 3 3 2 4 3 3 2 4 3 18 24 6 48 18 48 66 24 6 30 11 4 6 5 + = + × = + = = 2 3 2 3 4 2 2 3 2 3 4 2 3 4 2 3 4 2 6 17 2 24 9 32 30 17 2 23 30 17 2 23 + = + × + + = + + = + = + x x x x x x x x x x x x x x x x x x + + + = + ( ) + ( ) + ( )( ) ( ) + ( ) = + + ( ) ( ) + ( ) = + ( ) + ( ) 1 2 2 3 1 3 2 2 2 3 4 3 4 2 3 4 7 2 3 2 2 x y x y x y x y x y x y x y x y + ( )( ) ( ) ÷ + ( ) = + ( ) × + ( ) = + ( ) × 5 10 5 10 2 3 3 2 3 2 12 6 12 2 4 3 3 5 2 1 3 5 6 a a b a b a b a b b a ( ) ( ) [ ] = = 4 6 12 24 12 2 7 9 15 16 15 p p p p p p × = = Exercise 4.1 3. 4. b. ( ) = + ( ) ( ) = + ( ) + + ( ) + + + ( ) = + + = + = f f h f h h h h h h h h h h h h h h h 3 3 3 3 3 3 1 9 9 1 6 3 9 9 0 0 2 0 2 0 2 lim lim lim lim Now f h f h h h h f h h h f h 3 3 9 9 3 9 3 9 2 0 + ( ) ( ) = + = + ( ) ( ) = + ( ) ( ) = lim Since a f h h h h h f = + ( ) = + ( ) + + ( ) + = + + ( ) = + ( ) + = 3 3 3 3 3 1 9 19 3 3 3 3 1 19 2 2 2 , f x x x a f a f a h f a h h ( ) + + = ( ) = + ( ) ( ) 2 0 3 1 3 ; lim y x 1 1 y x Chapter 4: Derivatives 73 Chapter 4 • Derivatives

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c. ; 5. a. b. ( ) = + ( ) ( ) = + + + = + + + ( ) + ( ) = + + ( ) + ( ) ( ) = + ( ) f x f x h f x h x h x h x x h h x h x x h x f x x h h h h lim lim lim lim 0 0 0 0 2 3 2 3 2 3 6 3 3 6 2 2 3 2 2 3 2 f x x ( ) = + 3 2 ( ) = + f x x 2 3 ( ) = + ( ) ( ) = + ( ) + + ( ) + ( ) = + + = + + ( ) f x f x h f x h x h x h x x h hx h h h x h h h h h lim lim lim lim 0 0 2 2 0 2 0 3 3 2 3 2 3 f x x x ( ) = + 2 3 ( ) = + ( ) ( ) = + = + × + + + + = + + + ( ) = + + ( ) = f f h f h h h h h h h h h h h f h h h h h 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 2 0 0 0 0 0 lim lim lim lim lim a = 0 f x x ( ) = + 1 c. d. 6. b. dy dx x h x h x x h x xh x x h x xh x x h h x h x h h x h x dy dx x h h h = + + + + = + + + + + ( )( ) = + ( )( ) = ( ) lim lim lim 0 0 2 2 0 2 1 1 1 1 1 1 1 1 2 1 1 2 1 y x x = + 1 1 ( ) = + ( ) ( ) = + ( ) = + ( ) ( ) = + ( ) ( ) = ( ) = f x f x h f x h x h x h x x xh h h x h x x h x h x x x f x x h h h h lim lim lim lim 0 0 2 2 0 2 2 2 2 2 0 2 2 4 3 1 1 2 2 2 2 f x x ( ) = 1 2 f x x f x f x h f x h x h x h x h x x h x x h x h x h x f x x h h h ( ) = + ( ) = + ( ) ( ) = + + + × + + + + + + + + = + + + + + + ( ) ( ) = + 3 2 3 3 2 3 2 3 3 2 3 2 3 3 2 3 2 3 3 2 3 2 3 3 2 3 2 3 2 3 0 0 0 lim lim lim 2 74 Chapter 4: Derivatives
7. Since Slopes of the tangents at x = 0, 1, and 2 are , . 8. s ( t ) = – t 2 + 8 t; t = 0, t = 4, t = 6 s t h s t ht h h h t h + ( ) ( ) = + = + ( ) 2 8 2 8 2 s t h t h t h t ht h t h + ( ) = + ( ) + + ( ) = + + 2 2 2 8 2 8 8 v t s t s t h s t h h ( ) = ′ ( ) = + ( ) ( ) lim 0 y x 0 1 2 y = 2 x 2 – 4 x ( ) = ( ) = f f 1 0 2 4 , and ( ) = f 0 4 ( ) = + ( ) = + ( ) ( ) = f x h x h h x h f x x h h lim lim 0 0 4 2 4 4 2 4 4 4 f x f x h f x h f x h x h x h x hx h x h f x x x f x h f x hx h h h ( ) = + ( ) ( ) + ( ) = + ( ) + ( ) = + + ( ) = + ( ) ( ) = + lim 0 2 2 2 2 2 2 4 2 4 2 4 4 2 4 4 2 4 y f x x x = ( ) = 2 4 2 y x x = 2 4 2 Velocities at , 4, and 6 are and . 9. , parallel to The slope of the tangent to is parallel to . The point of tangency will be . The equation of the line will be or . x y – 6 10 0 + = y x 3 1 6 8 = ( ) 8 8 8 3 , , f ( ) ( ) = ( ) ( ) = + = + = = f x x x x 1 6 1 2 1 1 6 1 3 8 x y – 6 4 0 + = f x x ( ) = + 1 ( ) = + ( ) ( ) = + + + = + + + × + + + + + + + + = + + + ( ) + + + + ( ) = + + + + f x f x h f x h x h x h x h x h x h x x h x x h x h x h x x h x f x h h h h h lim lim lim lim lim 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( ) = + 1 2 1 x x y – 6 4 0 + = f x x ( ) = + 1 v

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ch04 - Chapter 4 Derivatives Review of Prerequisite Skills...

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