diff_tut_sol2 - Differentiation II Solutions to...

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Unformatted text preview: Differentiation II Solutions to Differentiation II Exercises. 1. f(x) = cos 2x f(0) = 1 f (x) = 2 sin 2x f (0) = 0 f (x) = 4 cos 2x f (0) = 2 2 f (x) = 8 sin 2x f (0) = 0 f (4) (x) = 16 cos 2x f (4) (0) = 2 4 ( 29 ( 29 ( 29 ( 29 ( 29 2 3 x x f x f 0 xf 0 f f ... 2! 3! = + + + + 2 2 4 4 6 6 2 x 2 x 2 x cos2x 1 ... 2! 4! 6! = - +- + 2. ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 n n n 1 n 1 n 2 n 2 n 3 n 2 f x 1 x f (0 ) 1 1 f x n 1 x f (0 ) n 1 n f x n n 1 1 x f (0 ) n n 1 1 n n 1 f x n n 1 n 2 1 x f (0 ) n n 1 n 2 1 n n 1 n 2------ = + = = = + = = = - + = - = - = - - + = - - = - - f (x) = n(n1)(n2)(1+x) n3 f (0) = n(n1)(n2)(1 n3 ) = n(n1)(n2) ( 29 ( 29 ( 29 ( 29 ( 29 2 3 x x f x f 0 xf 0 f f ... 2! 3! = + + + + ( 29 ( 29 ( 29 ( 29 2 3 n n n 1 x n n 1 n 2 x 1 x 1 nx ... 2! 3!--- + = + + + + 3. a) f(x) = cos 2 x f(0) = 1 f (x) = 2 cos x sin x f (0) = 0 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 ( 4 ) ( 4 ) ( 5 ) ( 5 ) ( 6 ) ( 6 ) f x co s x f 0 1 f x 2 co s x sin x f 0 1 f x sin 2 x f x 2 co s 2 x f 0 2 f x 4 sin 2 x f 0 0 f x 8 co s 2 x f 0 8 f x 1 6 sin 2 x f 0 0 f x 3 2 co s 2 x f 0 3 2 = = = - = = - = - = - = = = = = - = = - = - ( 29 ( 29 ( 29 ( 29 ( 29 2 3 x x f x f 0 xf 0 f f ... 2! 3! = + + + + 2 4 6 2 4 6 2 2 2x 8x 32x cos x 1 .... 2 24 720 x 2x cos x 1 x .... 3 45 = - +- + = - +- + 1 Differentiation II b) x xlna x xln2 now a e 2 e = = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 xln2 xln2 2 2 xln2 3 3 xln2 f x e f 0 e 1 f x ln 2e f 0 ln 2 f x ln 2 e f ln 2 f x ln 2 e f ln 2 = = = = = = = = = ( 29 ( 29 ( 29 ( 29 ( 29 2 3 x x f x f 0 xf 0 f f ... 2! 3!...
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This note was uploaded on 04/25/2010 for the course MATH 201 taught by Professor Any during the Spring '10 term at Westminster UT.

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diff_tut_sol2 - Differentiation II Solutions to...

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