HM4-soln - Homework 4 Solution October 17, 2011 6.47 1. x 2...

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Unformatted text preview: Homework 4 Solution October 17, 2011 6.47 1. x 2 + y 2 6 = 1 . 1 2. x + y > . 3. 2 x- y x + y 1 . 2 6.59 (a) Continuous. The function is the summation of two continuous func- tions. (b) Discontinuous. Since lim x (lim y x 3 x + 5 y ) = lim x 1 3 = 1 3 , lim y (lim x x 3 x + 5 y ) = lim y 0 = 0 , lim ( x,y ) (0 , 0) x 3 x +5 y cannot exist. (c) Continuous. lim ( x,y ) (0 , 0) ( x 2 + y 2 )sin 1 x 2 + y 2 = 0 . 6.63 (a) f x (0 , 0) = lim h f ( h, 0)- f (0 , 0) h = lim h ( h 2- 0) / ( h + 0) h = 1 . (b) f y (0 , 0) = lim h f (0 ,h )- f (0 , 0) h = lim h (0- 0) / (0 + h ) h = 0 . 6.67 (a) f x = 2( x + y )- (2 x- y ) ( x + y ) 2 = 3 y ( x + y ) 2 , f xy = 3( x + y ) 2- 3 y 2( x + y ) ( x + y ) 4 = 3( x- y ) ( x + y ) 3 . f y =- 3 x ( x + y ) 2 , f yx = 3( x- y ) ( x + y ) 3 = f xy . Exceptional points x + y = 0 . Derivatives dont exist. (b) f x = tan xy + xy sec 2 xy, f xy = sec 2 xy x + x sec 2 xy + xy 2sec xy sec xy tan xy x = 2 x sec 2 xy +2 x 2 y sec 2 xy tan xy. f y = x sec 2 xy x = x 2 sec 2 xy, f yx = 2 x sec 2 xy + x 2 2sec xy sec xy tan xy y = 2 x sec 2 xy +2 x 2 y sec 2 xy tan xy = f xy . 3 Exceptional points xy = k + 1 2 ,k = 0 , 1 , 2 ,.... Derivatives dont exist. (c) f x = sinh( y + cos x ) (- sin x ) =- sin x sinh( y + cos x ) , f xy =- sin x cosh( y + cos x ) . f y = sinh( y + cos x ) , f yx = cosh( y + cos x ) (- sin x ) = f xy . No exceptional points. 6.68 z x = 2( x- a ) ( x- a ) 2 + ( y- b ) 2 . z xx = 2[( x- a ) 2 + ( y- b ) 2 ]- 2( x- a ) 2( x- a ) [( x- a ) 2 + ( y- b ) 2 ] 2 = 2[( y- b ) 2- ( x- a ) 2 ] [( x- a ) 2 + ( y- b ) 2 ] 2 . Similarly we have z yy = 2[( x- a ) 2- ( y- b ) 2 ] [( x- a ) 2 + ( y- b ) 2 ] 2 . It is clear that z xx + z yy = 0 . 6.69 z x = cos y x + y x sin y x- y x 2 sec 2 y x , z y =- sin y x + 1 x sec 2 y x , z xx =- y 2 x 3 cos y x + 2 y x 3 sec 2 y x + 2 y 2 x 4 tan y x sec 2 y x , z xy = y x 2 cos y x- 1 x 2 sec 2 y x- 2 y x 3 sec 2 y x tan y x , z yy =- 1 x cos y x + 1 x 2sec y x...
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This note was uploaded on 02/28/2012 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.

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HM4-soln - Homework 4 Solution October 17, 2011 6.47 1. x 2...

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