x2a_sols - Fall 2010 Math 251 Exam 2A: Solutions Tue,...

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Unformatted text preview: Fall 2010 Math 251 Exam 2A: Solutions Tue, 19/Oct c 2010 Art Belmonte 1. Describe level curves of f ( x , y ) = p 25- x 2- y 2 . Sketch the level curves for the c-values c = , 3 , 4 on the grid below.-5 5-5 5 x y Problem 1 Level curves are where function values equal constants. f ( x , y ) = c p 25- x 2- y 2 = c 25- x 2- y 2 = c 2 x 2 + y 2 = 25- c 2 = r 2 These are circles . For c = , 3 , 4, we have x 2 + y 2 = 25 , 16 , 9, respectively. These are circles of radii 5 , 4 , 3, respectively, centered at the origin. They are drawn on the grid above. 2. For each part, find the limit if it exists or show why it does not exist. (a) lim ( x , y ) ( , ) x 2 y x 4 + 4 y 2 As ( x , y ) ( , ) along the x-axis ( y = 0), we have x 2 y x 4 + 4 y 2 = x 4 = 0. As ( x , y ) ( , ) along the parabola y = x 2 , we have x 2 y x 4 + 4 y 2 = x 4 5 x 4 = 1 5 1 5 . Since these two directional limits differ, the multivariable limit lim ( x , y ) ( , ) x 2 y x 4 + 4 y 2 does NOT exist. (b) lim ( x , y ) ( , 1 ) tan- 1 x 2 + 1 x 2 +( y- 1 ) 2 ! As ( x , y ) ( , 1 ) , we have x 2 + 1 x 2 +( y- 1 ) 2 + , whence tan- 1 x 2 + 1 x 2 +( y- 1 ) 2 ! 2 . Therefore, lim ( x , y ) ( , 1 ) tan- 1 x 2 + 1 x 2 +( y- 1 ) 2 !...
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x2a_sols - Fall 2010 Math 251 Exam 2A: Solutions Tue,...

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