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a1_soln

a1_soln - Math 237 Assignment 1 Solutions 1 Let f x y =...

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Unformatted text preview: Math 237 Assignment 1 Solutions 1. Let f ( x, y ) = radicalbig 4 + x 2- y 2 . i) Sketch the domain of f and state the range of f . Solution: The domain of f is 4 + x 2- y 2 ≥ ⇒ x 2- y 2 ≥ - 4. The range is z ≥ 0. ii) Sketch level curves and cross sections. Solution: Level Curves: k = radicalbig 4 + x 2- y 2 x 2- y 2 = k 2- 4, k ≥ Cross sections: z = radicalbig 4 + c 2- y 2 z 2 + y 2 = 4 + c 2 , z ≥ z = √ 4 + x 2- d 2 z 2- x 2 = 4- d 2 , z ≥ 2. Find the limit, if it exists, or show that the limit does not exist. a) lim ( x,y ) → (0 , 0) x 4 +2 y 4 x 4 + y 4 Solution: Approaching the limit along lines y = mx we get lim ( x,y ) → (0 , 0) x 4 + 2 m 4 x 4 x 4 + m 4 x 4 = lim x → 1 + 2 m 4 1 + m 4 = 1 + 2 m 4 1 + m 4 . Hence, since we get a different limit for each value of m the limit does not exist. 1 2 b) lim ( x,y ) → (0 , 0) ( x- y ) 2 | x | + | y | Solution: Approaching the limit along lines y = mx we get lim ( x,y, ) → (0 , 0) ( x- mx ) 2 | x | + | mx | = lim...
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a1_soln - Math 237 Assignment 1 Solutions 1 Let f x y =...

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