{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows page 1. Sign up to view the full content.

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 (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 4. For F ( x, y, z
This is the end of the preview. Sign up to access the rest of the document.

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 it’s 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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online