assignment2sol - ma2176 a2 sol Derivatives 1. If y = x sin...

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1 ma2176 a2 sol Derivatives 1. If sin yx x = , prove that ( ) 22 '' 2 ' 2 0 xy x y x y −+ + = . Proof : sin ' sin cos " cos cos sin 2cos sin yx x y xx x y x xx x =⇒ = +⇒ = + −=− . So () ( ) ( ) ( ) ( ) 2 2 32 '' 2 ' (2 ) sin 2 sin cos 2 sin sin 2sin 0 x yx y x y x x x xx x x x x x xxx + = + + + =− + + + + = 2. If c bx ax u + + = 2 2 , prove that u c bx ax xu dx d + + = 3 2 ) ( 2 . Proof : ( ) 11 222 1 2 2 1 2( 2 ) ( 2 ) ( 2 2 ) 2 (2) du d d ax b ax b ax bx c ax bx c ax bx c ax b dx dx dx u ax bx c  ++ =+ + = + + = + + + = =   Then 2 2 2 3 d du ax b ax bx u ax bx ax bx c ax bx c xu x u x u dx dx u u u u + + + + + + + = + = = = . 3. Differentiate the following functions with respect to x : (a) 3000 2 12 5 +− , (b) ( ) ( ) 33 sin 2 5cos 1 + , (c) ln ln ln x . Solution : (a) 3000 2999 1 2 5 3000 1 2 5 2 10 d x x x dx = (b) () () ( ) ( ) 2 3 2 23 2 2 2 3 sin 2 5 cos 1 3 sin 2 cos 2 (2) 5 sin 1 3 3 sin 2 cos2 (2) 5sin 1 3 6sin 2 cos2 15 sin 1 d x x x x dx x x x x = + + = + + (c) [] ln 1 ln(ln ) 1 ln ln ln(ln ) ln ln ln ln ln ln ln ln ln ln d x d dx x d dx x x x dx x x x x x x x ==   4. Find the values of a and b (in terms of c ) such that ) ( ' c f exists, where + > = c x b ax c x |x| x f | | if | | if 1 ) ( . Solution :
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2 < + > = + > = c x x c x c b ax c x x x f c x b ax c x |x| x f if 1 if if 1 ) ( | | if | | if 1 ) ( In order that ) ( ' c f exists, ) ( x f must be continuous at c x = . That is, ) ( ) ( lim 1 1 lim ) ( lim c f b ac x f c x x f c x c x c x = + = = = = + + . 2 11 1 () 1 1 lim lim lim lim lim lim xc cx ac b fx fc xx c x c x c x c x c x c x c x c c + + + +++ →→→ −+ == = = = = −− . ( ) ( ) lim lim lim a x b a c b ax c a →→ + = . That ) ( ' c f exists means a c x c f x f c c x c f x f c x c x = = = + ) ( ) ( lim 1 ) ( ) ( lim 2 .
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This note was uploaded on 11/09/2009 for the course ENG 91301 taught by Professor Lui during the Spring '08 term at Hong Kong Institute of Vocational Education.

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assignment2sol - ma2176 a2 sol Derivatives 1. If y = x sin...

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