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math20c Meesue Yoo fall 09 quiz4

math20c Meesue Yoo fall 09 quiz4 - Quiz 4 Solution for MATH...

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Quiz 4 Solution for MATH 20C, 2009 1 . Evaluate the limit or determine that the limit does not exist. lim ( x,y ) (0 , 0) y 2 x 2 + y 2 Ans. We choose two different path approaching (0 , 0) and find that there exist two limit values along different paths. First, along the y -axis, i.e., when x = 0, lim (0 ,y ) (0 , 0) y 2 y 2 = lim (0 ,y ) (0 , 0) 1 = 1 . Secondly, if we choose the x -axis, i.e., when y = 0, lim ( x, 0) (0 , 0) 0 x 2 + 0 = lim ( x, 0) (0 , 0) 0 = 0 . We checked that along two different coordinate axes, the limit values are different. Hence, the limit does not exist . 2 . Compute the derivative indicated. f ( x, y ) = x ln( y 2 ) , f yy (2 , 3) Ans. Note that f ( x, y ) = 2 x ln y . Then the first and second partial derivatives of f with respect to y are f y ( x, y ) = 2 x y , f yy ( x, y ) = - 2 x y 2 f yy (2 , 3) = - 4 9 3 . Use linear approximation to
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