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010-104-2002-2-mid - > 5 o 2 cfw 1r 3r 4 Z 9 < s2 j h...

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2 2002 10 26 1 – 3 : : ( 100 ). 1. (10 ) Consider the region bounded by the curve y = x + 1, the x - axis, and the y -axis. Let V x , V y be the volumes of the solid generated by revolving the region about the x -axis and the y -axis, respectively. Find the ratio V x : V y (Write it as simple as possible). 2. (15 ) f ( x, y ) = x 2 y 3 x 2 + 4 y 3 , ( x, y ) = (0 , 0) 0 , ( x, y ) = (0 , 0) . (a) Find f x (0 , 0) and f y (0 , 0). (b) Prove that f is not continuous at the origin. 3. (10 ) Find the maxima, minima, and saddle points of f ( x, y ). f ( x, y ) = sin x + sin y + sin( x + y ) . 4. (10 ) Find the points on the curve xy +2 xz = 5 5 nearest to the origin. 5. (10 ) Let (0 , 0) be a critical point of a differentiable function z = f ( x, y ) . Does the condition “ f xx > 0 and f yy > 0” imply that f has a local minimum at (0 , 0)? If not, find a counter example and a condition which
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