a1_soln - Math 237 Assignment 1 Due: Friday, Sept 25th 1....

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Unformatted text preview: Math 237 Assignment 1 Due: Friday, Sept 25th 1. For each of the following functions 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 . Solution: The domain is R 2 and the range is z 0. Level Curves: k = x 2 Cross sections: z = c 2 z = x 2 Thus, the surface looks like 1 2 b) f ( x, y ) = radicalbig | 1- x 2- y 2 | . Solution: The domain is R 2 and the range is z 0. Level Curves: k = radicalbig | 1- x 2- y 2 | x 2 + y 2 = 1 k 2 , k Cross sections: z = radicalbig | 1- c 2- y 2 | y 2 z 2 = 1- c 2 , z z = radicalbig | 1- x 2- d 2 | x 2 z 2 = 1- d 2 , z Thus, the surface looks like 2. Find the limit, if it exists, or show that the limit does not exist. a) lim ( x,y ) (0 , 0) x 2- y 2 x 2 + y 2 . Solution: Approaching the limit along lines y = mx we get lim ( x,y ) (0 , 0) x 2- m 2 x 2 x 2 + m 2 x 2 = lim x 1- m 2 1 + m 2 = 1- m 2 1 +...
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a1_soln - Math 237 Assignment 1 Due: Friday, Sept 25th 1....

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