SecFEx2W01 - 9 For the function f x y =(ln y 2 x(a Find the...

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M.Thomas Winter 2001 Math 22 Exam#2 1. Find the following limit, if it exists, or show that it does not exist. lim ( x,y,z ) (0 , 0 , 0) xy 2 + yz + zx x 6 + y 2 +3 z 2 2. Draw a contour map of the function f ( x, y ) = x + y x - y showing several level curves. 3. Given xyz 2 + xy 2 + xz 3 = x - y + zx , find ∂x ∂y . 4. If W ( s, t ) = F ( u ( s, t ) , v ( s, t )) where u (2 , - 1) = 3, u s (2 , - 1) = 1, u t (2 , - 1) = 4, v (2 , - 1) = 5, v s (2 , - 1) = 2, v t (2 , - 1) = - 1, F u (3 , 5) = 7, and F v (3 , 5) = 6, find W s (2 , - 1). 5. Show whether or not the function f ( x, y ) = 1 6 x 3 - 1 2 xy 2 + xy satisfies Laplace’s equation. 6. If f ( x, y, z ) = sin( xy + yz ), find f yxx . 7. For the function g ( s, t ) = t 2 e s find (a) the directional derivative in the direction of v = 5 i - 12 j at the point (0 , - 3). (b) the maximum rate of change of g at the point (0 , - 3) and the direction in which it occurs 8. Find the equation of the normal line to the surface x 2 +4 y 2 - z 2 = - 1 at the point (2
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Unformatted text preview: 9. For the function f ( x, y ) = (ln( y )) 2 x (a) Find the linearization L ( x, y ) at the point (1 , e 2 ). (b) Determine if f ( x, y ) is differentiable over its domain. 10. A 3 in. wide boundary stripe is painted around the perimeter of a rectangular field whose dimensions are 150 ft. by 300 ft. Use differentials to approximate the number of square feet of paint in the stripe. 11. For the function f ( x, y ) = x 3-yx + y 3 (a) Find its local maximums, local minimums, and all of its saddle points. (b) Explain why it is or is not possible to find any absolute maximums or absolute mini-mums for f in the region R = [( x, y ) : | x | + | y | < 1]....
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