# tut7 - f x y and describe the shape of the level curves...

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Math 237 Tutorial 7 Problems 1: Let F : R 2 R 2 given by ( u, v ) = F ( x, y ) = ( y + e x , y - e x ). Sketch the image of the square with vertices (1 , 0), (0 , 1), (0 , 0), and (1 , 1), under F . 2: Let F : R 2 R 2 be deﬁned by F ( u, v ) = ( ve u , u + v ) and G : R 2 R 2 be deﬁned by G ( x, y ) = ( x 2 y, x + 2 y ). a) Find ( F G )( x, y ) and compute D ( F G ) using the chain rule in matrix form. b) Calculate D ( G F )(2 , 1). 3: Find the image of the circle x 2 + y 2 = 1, under the mapping F ( x, y ) = ± p x 2 + y 2 , tan - 1 ( y/x ) ² . 4: Let f ( x, y ) = x 3 + y 3 - 3 x - 3 y + 2. a) Find and classify the critical points of
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Unformatted text preview: f ( x, y ) and describe the shape of the level curves near the critical points. b) Find the maximum and minimum of f ( x, y ) on the unit disc x 2 + y 2 ≤ 1. 5: Consider the two curves in polar coordinates r = θ and r = π 2 . Sketch both curves on the same axis and ﬁnd the area inside both curves....
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## This note was uploaded on 04/18/2010 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.

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