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Quiz #3
NAME:
1. (8 pts) Let f ( x, y)  ( xy 2  3x)3
a. Find f x ( x, y ), f y ( x, y ), and f xy ( x, y )
b. Evaluate f (1, 2) .
c. Find the equation of the tangent plane to the graph of f at (1, 2)
2.
(8 pts) Evaluate each of the following limits, or show that it does not exist.
 x2

a. Lim 

 x , y    0,0   x 2  y 2 


b.
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Lim
 x , y    0,0
 x3 y 
 x6  y 2 


3.
 x2  4 y 2
if ( x, y)  (0,0)

(5 pts) Let G( x, y)   y ( x  2 y )
.
 0
if ( x, y )  (0,0)

Determine with explanation if G is continuous at (0,0) .
4.
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(5 pts) Sketch the graph of z 2  x 2  y 2  1 .
5. (5 pts) Suppose that you have a function f :
( x, y )
(1, 2)
(1.05, 2)
(1,1.99)
2

with table of values given below:
f ( x, y )
4
4.1
4.01
a. Estimate the values of f x (1,2) and f y (1, 2) .
b. Give an approximation for the tangent plane to the graph of f at (1,2).
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6.
 2
 1 
2
if ( x, y )  (0,0)
 x  y  sin  2
2 
(3 pts) Let f ( x, y )  
x y 

0
if ( x, y )  (0,0)

a. Determine f x (0,0) or show that it does not exist.
b. is f x continuous at (0,0)? Explain/show work.
c. Determine whether f is differentiable at (0,0) . Again, Explain/show work.
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