diff_tut_sol1 - Differentiation I Solutions to...

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Differentiation I Solutions to Differentiation I Exercises. 1. (i) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 h 0 h 0 1 f x x 3 1 f x h x h 3 1 1 f x h 3 x 3 x 3 x h 3 f x h 3 x 3 x 3 x h 3 x 3 x h 3 f x h 3 x 3 x 3 x h 3 x 3 x h 3 f x h 3 x 3 x 3 x h 3 h f x h 3 x 3 x 3 x h 3 f 1 now f (x) lim lim h x h 3 x 3 x 3 x h 3 1 f x x 3 x 3 x 3 = + + = + + ∆ = - + + + + - + + ∆ = + + + + - + + + + + + ∆ = + + + + + + + + - + + ∆ = + + + + + + + - ∆ = + + + + + + + - = = + + + + + + + - = + + + ( 29 ( 29 ( 29 3 x 3 1 f x 2 x 3 + + - = + (ii) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 x f (x) 1 x x h f (x h) 1 x h x h x f 1 x h 1 x x h 1 x x 1 x h f 1 x h 1 x x h x xh x x xh f 1 x h 1 x = - + + = - - + ∆= - - - - + + - - - ∆= - - - + - - - + + ∆= - - - ) x 1 )( h x 1 ( xh x x xh x h x ) x 1 )( h x 1 ( ) h x 1 ( x ) x 1 )( h x ( f 2 2 - - - + + - - - + = - - - - - - - + = ( 29 ( 29 ( 29 ( 29 ( 29 h 0 h 0 2 h f 1 x h 1 x f 1 Now f (x) lim lim h 1 x h 1 x 1 f (x) 1 x ∆ = - - - = = - - - = - 1
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Differentiation I 2. (i) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 { } ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 x 1 2x 5 f (x) 2x 3 let u x 1 2x 5 and v 2x 3 u x v x u x v x f x v x 2 x 1 2x 5 2x 3 2 x 1 2x 5 f x 2x 3 4x 3 2x 3 2 x 1 2x 5 f x 2x 3 8x 6x 12x 9 4x 4x 10x 10 f x 2x 3 4x 12x 1 f x 2x 3 + - = + = + - = + - = + + - + - + - = + - + - + - = + - + - - - + + = + + + = + (ii) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 2 2 2 x 1 f (x) x 2x 2 let u x 1 and v x 2x 2 u x v x u x v x f x v x x 2x 2 x 1 2x 2 f x x 2x 2 x 2x 2 2x 2x 2x 2 f x x 2x 2 x 2x f x x 2x 2 + = + + = + = + + - = + + - + + = + + + + - - - - = + + + = - + + (iii) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 10 9 9 f x 3x 5 f x 10 3x 5 3 f x 30 3x 5 = + = + = + (iv) 3 1 2 2 2 ) x x ( ) 2 x ( ) x ( f - + + = let u = (x 2 + x –1 ) 3 and v = (x 2 + 2) 2 2
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Differentiation I ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 { } ( 29 ( 29 ( 29 2 2 3 2 1 2 2 3 2 2 2 2 1 2 2 1 2 6 2 1 2 2 2 1 2 2 4 2 1 2 2 3 2 2 4 2 1 2 2 5 2 2 x 2 f (x) x x let u x 1 and v x 2x 2 u x v x u x v x f x v x 2 x 2 2x x x 3 x 2 x x 2x x f (x) x x 4x x 2 x x 3 x 2 2x x f (x) x x x 2 4x 4 3 x 2 2x x f (x) x x x 2 4x 4x 3 x 2 2 f (x) - - - - - - - - - - + = + =+ =++ ′ ′ - ′ = + + - + + - ′ = + + + - + - ′ = + + +- + - ′ = + + + - + ′ =
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diff_tut_sol1 - Differentiation I Solutions to...

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