quiz2 - 2 + 2 z 2-yz + y decide if it is possible to solve...

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MAT 200. Spring 2004. Quiz 2. April 8, 2004 1. Compute the directional derivative of the function f ( x, y, z ) = ln( e x + e y + e z ) at point (0 , 0 , 0) in the direction ¯ v = (1 , 4 , 1) . Solution: f ( x, y, z ) = 1 e x + e y + e z ( e x , e y , e z ) , so f (0 , 0 , 0) = 1 3 (1 , 1 , 1) and D ¯ v f (0 , 0 , 0) = f (0 , 0 , 0) · ¯ v = 2 . 2. Compute the derivative matrix of the function F ( x, y ) = (( x - y 3 ) e sin 3 ( x + y ) - 2 cos 2 ( x + y ) , ln( x - y 3 ) + sin( x + y )) at point (1 , - 1) . DF (1 , - 1) = ± 1 - 3 3 / 2 - 1 / 2 ² (use the following change of variables: ( u, v ) = ( x - y 3 , sin( x + y ))) 3. Compute the Hessian matrix of the function f ( x, y ) = e x - y ( x 2 - 2 y 2 ) at point (2 , 1) . H (2 , 1) = ± 12 e - 10 e - 10 e 6 ² 4. For F ( x, y, z ) = x 2 - 2 y
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Unformatted text preview: 2 + 2 z 2-yz + y decide if it is possible to solve the equation F ( x, y, z ) = 0 for z in terms of x and y in a neighborhood of the point (4 , 3 , 1) . If yes, estimate z (4 . 125 , 2 . 75) . Solution: Yes, F z = 4 z-y so its non-zero at (4 , 3 , 1) . z x =-2 x 4 z-y , z y =-1-4 y-z 4 z-y , and linear approximation gives z (4 . 125 , 2 . 75) = 1+(-8)( . 125)+ 12(-. 25) =-3 . 1...
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