Points determine whether the following function is

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Problem 10 (16 points) Determine whether the following function is continuous at the point ( 0, 0 ) . Show all your work.
Problem 11 (16 points) Consider the surface x 2 + y 2 + z 2 + 3 xyz = 0. Find an equation for the tangent plane to the surface at the point ( 2, - 1, 1 )
Problem 12 (16 points) Find all critical points of the function f ( x , y ) = x 2 - 2 xy + y 3 and determine whether each is a local maximum, local minimum, or a saddle point.
Problem 13 (16 points) Find the point(s) on the cirlce x 2 + y 2 = 1 that are closest to the point P = ( 2, 1 ) .
Problem 14 (14 points) A differentiable function f has f ( 1, 0 ) = 1, f x ( 1, 0 ) = 2, f y ( 1, 0 ) = 3, f xx ( 1, 0 ) = 4, f xy ( 1, 0 ) = 5 and f yy ( 1, 0 ) = 6. (a) (4 points) Write the equation of the tangent plane to the surface z = f ( x , y ) at the point ( 1, 0 ) . (b) (4 points) Use a linear approximation to estimate f ( 0.8, 0.3 ) . (c) (6 points) Find the second order Taylor polynomial (quadratic approximation) of f about ( 1, 0 ) .

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