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A1_SOLN - Math 237 Assignment 1 Due Friday Sept 19th 1 For...

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Math 237 Assignment 1 Due: Friday, Sept 19th 1. For each of the following function f : R 2 R i) Sketch the domain of f and state the range of f . ii) Sketch level curves and cross sections. iii) Sketch the surface z = f ( x,y ). a) f ( x,y ) = x 2 - y 2 + 1. Solution: i) The domain is R 2 the range is z R . ii) Level Curves: k = x 2 - y 2 + 1 x 2 - y 2 = k - 1 Cross sections: z = c 2 - y 2 + 1 z = x 2 - d 2 + 1 iii) 1
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2 b) f ( x,y ) = radicalbig x 2 + y 2 Solution: i) The domain is R 2 the range is z [0 , ). ii) Level Curves: k = radicalbig x 2 + y 2 x 2 + y 2 = k 2 , k 0 Cross sections: z = radicalbig c 2 + y 2 z 2 - y 2 = c 2 , z 0 z = x 2 + d 2 z 2 - x 2 = d 2 , z 0 iii)
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3 2. Find the limit, if it exists, or show that the limit does not exist. a) lim ( x,y ) (0 , 0) x 4 + y 4 x 2 + y 2 Solution: Try approaching the limit along lines y = mx . This gives lim ( x,y ) (0 , 0) x 4 + y 4 x 2 + y 2 = lim ( x,y ) (0 , 0) x 4 + m 4 x 4 x 2 + m 2 x 2 = lim x 0 x 2 + m 4 x 2 1 + m 2 = 0 Since all these limits give the same value, the limit may be equal to 0. We will try to prove it using the squeeze theorem. Observe that x 4
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