mid1bsol - Problem 1 Find the limits below or explain why...

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Problem 1. Find the limits below or explain why they do not exist: (i) lim x,y 0 ( x 2 + y 2 ) 2 2 x 2 +3 y 2 We note that 0 ( x 2 + y 2 ) 2 2 x 2 + 3 y 2 (2 x 2 + 3 y 2 ) 2 2 x 2 + 3 y 2 = 2 x 2 + 3 y 2 0 . Therefore, the original limit equals 0 as well. (ii) lim x,y 0 x 3 y x 4 + y 4 The limit does not exist. Indeed, approaching 0 by keeping x = y 0 , the fraction equals x 3 · x x 4 + x 4 = 1 2 . On the other hand, approaching 0 by keeping x = 0 , y 0 , the fraction equals 0 . Since the two answers are different, the limit does not exist.
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Problem 2 Consider the function f ( x, y ) = x 2 sin(2 y - 2 x ) and the point P (1 , π 2 + 1) . (i) Find the gradient of f at the point P . We compute the derivatives f x = 2 x sin(2 y - 2 x ) - 2 x 2 cos(2 y - 2 x ) = f x (1 , π 2 + 1) = 2 sin π - 2 cos π = 2 f y = 2 x 2 cos(2 y - 2 x ) = f y (1 , π 2 + 1) = 2 cos π = - 2 . Then f ( P ) = (2 , - 2) . (ii) Calculate the directional derivative of f at P in the direction ~u = ~ 3 i + 4 ~ j 5 . We have f ~u ( P ) = f ( P ) · ~u = (2 , - 2) · 3 5 , 4 5 = 2 · 3 5 - 2 · 4 5 = - 2 5 . (iii) Find the (unit) direction of steepest decrease for the function f ( x, y ) at P . What is the rate of de- crease?
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