Practice Exam 2 - xyz at the point (1 ; 2 ; 1) 4. Let f (...

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Warning: The following is a practice exam. Do not expect the actual exam to necessarily consist of the same or similar problems. 1 (a) Find the limit, or show it does not exist: lim ( x;y ) ! (0 ; 0) x 2 y 2 p x 4 + y 4 : (b) Find the limit, or show it does not exist: lim ( x;y ) ! (0 ; 0) x 2 y 2 x 4 + y 4 . 2. Let f ( x; y ) = 4 p x 4 + 4 xy + y 4 ; if ( x; y ) 6 = (0 ; 0) 0 ; if ( x; y ) = (0 ; 0) (a) Determine f x (0 ; 0) , or show it does not exist. (b) Determine whether f is di/erentiable at (0 ; 0) , justifying your conclusion. 3. Suppose x 3 + y 3 + z 3 + 2 = 6 xyz z implicitly as a function of x and y . (a) Determine @z @x and @z @y . (b) Determine the equation of the tangent plane to the graph of x 3 + y 3 + z 3 + 2 = 6
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Unformatted text preview: xyz at the point (1 ; 2 ; 1) 4. Let f ( x; y ) = xy 1 + x 2 + y 2 (a) At the point (1 ; 2) , &nd the direction vector u which maximizes the directional derivative ( D u f ) (1 ; 2) : (b) Determine the maximal value of ( D u f ) (1 ; 2) . 5. Use the second derivative test to determine all local maximum, local minimum, and saddle points of f ( x; y ) = xye & x 2 = 2 & y 2 = 8 . 6. Find the maximum and minimum of f ( x; y; z ) = x & y + z subject to the constraint x 2 +2 y 2 + 3 z 2 = 1 ....
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