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a3_soln

# a3_soln - Math 237 Assignment 3 Solutions 1 Let f x,y = xy...

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Unformatted text preview: Math 237 Assignment 3 Solutions 1. Let f ( x,y ) = xy x 2 + y 2 . a) Find the equation of the tangent plane of f at (1 , 2 , 2 / 5). Solution: We have f x = y ( x 2 + y 2 )- 2 x 2 y ( x 2 + y 2 ) 2 = y 3- x 2 y ( x 2 + y 2 ) 2 f y = x 3- xy 2 ( x 2 + y 2 ) 2 Thus, the equation of the tangent plane is z = f (1 , 2) + f x (1 , 2)( x- 1) + f y (1 , 2)( y- 2) = 2 5 + 6 25 ( x- 1)- 3 25 ( y- 2) b) Approximate f (0 . 9 , 2 . 1). Solution: f (0 . 9 , 2 . 1) ≈ L (1 , 2) (0 . 9 , 2 . 1) = 2 5 + 6 25 (0 . 9- 1)- 3 25 (2 . 1- 2) = 0 . 4- . 024- . 012 = 0 . 364 2. For each of the following functions f : R 2 → R , determine if f is differentiable at (0 , 0). a) f ( x,y ) = ( x 3 + y 4 x 2 + y 2 if ( x,y ) 6 = (0 , 0) if ( x,y ) = (0 , 0) . Solution: We have f x (0 , 0) = lim h → f ( h, 0)- f (0 , 0) h = lim h → h 3 /h 2 h = 1 f y (0 , 0) = lim h → f (0 ,h )- f (0 , 0) h = lim h → h 4 /h 2 h = 0 , Thus, the error in the linear approximation is R 1 , (0 , 0) ( x,y ) = f ( x,y )- f (0 , 0)- f x (0 , 0)( x- 0)- f y (0 , 0)( y- 0) = x 3 + y 4 x 2 + y 2- x = y 2...
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a3_soln - Math 237 Assignment 3 Solutions 1 Let f x,y = xy...

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