practiceMID2

practiceMID2 - f ( x, y ) = ln ( x 2 + y 2 + 1). f ( x, y )...

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Math 10A, Winter 2009, UC Riverside Practice midterm 2 Dr. Mohamed Ait Nouh Handout (1) Compute the second partial derivative of f ( x, y ) = x 2 y + xy 2 + yz 2 . (2) Compute the matrix of partial derivatives of f ( x ) = (3 x + 2 y, 4 x 5 y, x + y + z ) (3) If f ( x, y ) = sin ( xy ) and x = s + t , y = s 2 + t 2 , Fnd ∂f ∂s and ∂f ∂t . (4) Compute the tangent plane at (1 , 0 , 1) for the function f ( x, y, z ) = x 2 y 3 + z 7 . (5) Compute the gradient f for the function f ( x, y, z ) = x 2 + y 3 + z 4 (6) Compute the directional derivative of f ( x, y, z ) = z 2 x + y 3 at (1 , 1 , 2) in the direction 1 5 i + 2 5 j . (7) ±ind the plane tangent to the surface x 2 + 2 y 2 + 3 xy = 10, at the point (1 , 2 , 1 3 ). (8) Compute the tangent plane at (1 , 0 , 1) for the function f ( x, y, z ) = x 2 y 3 + z 7 . (9) ±ind the local maxima, minima and saddle points for
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Unformatted text preview: f ( x, y ) = ln ( x 2 + y 2 + 1). f ( x, y ) = x 5 y + xy 5 + xy . (10) Compute the second order Taylor formula for f ( x ) = sinx and deduce lim x m sinx x = (11) Compute the second order Taylor formula for f ( x, y ) = e x 3 + y 2 at ( x , y ) = (0 , 0). (12) Evaluate the limits or explain why the limit fails to exist. lim ( x,y ) m (0 , 0) x 2 x 2 + y 2 = lim ( x,y ) m (0 , 0) x x 2 + y 2 = lim ( x,y ) m (0 , 0) sin ( x + y ) x + y = 1...
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