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SHOW ALL WORK !!
Quiz #3
NAME:
 y
1. (8 pts) Let f ( x, y )  sin x
a. Find f x ( x, y ), f y ( x, y ), and f xy ( x, y )
b. Evaluate f (1,  ) .
c. Find the equation of the tangent plane to the graph of f at (1,  ) .
2.(8 pts) Evaluate each of the following limits, or show that it does not exist.
 xy 2 
a. Lim  2
 x , y    0,0 x  y 4 


b.
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Lim
 x , y    0,0 


x


 x 2  y 2 
3. (5 pts) Let G( x, y)  3 x  y . Find Gx (0, 0) if it exists.
4. (5 pts) Sketch the graph of x 2  y 2  z 2  1 .
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5. (5 pts) Use the level curves for the function z  f ( x, y ) shown below to estimate
f (2, 2), f x (2, 2) and f y (2, 2). Show enough work so that I can follow your reasoning.
30
20
10
0
-10
-20
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6. (3 pts)
Suppose that
  fg   f g  g f
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f:
2

and g :
2

are differentiable. Prove that