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Chapter 2 Quiz
NAME:
#1. (6 pts) Find the equation of the tangent plane to the surface with equation z 3  zy  x 2e xy  5 at
the point (2, 0,1) .
#2. (6 pts) Find the directional derivative of F ( x, y)  sin( x 2  y 2 ) in the direction of the vector
v  i  j , at the point


,  .
#3. (5 pts) Write a formula for
z
if z  f ( x, y ) and x  x(r , s, t ), y  y (r , s, t ).
s
#4. (8 pts)
a.
b.
Evaluate each of the following limits, or show that it does not exist.
lim
 x , y    0,0 
lim
 e y sin  2 x  


x


( x , y ) (0,0)
 x3 y  x 2  y 2 
 x2  y 2 


#5(6 pt’s) Prove that if z  f ( x, y ) , where x  r cos  and y  r sin  , then
2
1  z 
 z   z   z 
       2

r   
 x   y   r 
2
2
 y  x2
if ( x, y )  (0, 0)

#6. (5 pts) Find f x (0, 0) if f ( x, y )   x
 0
if ( x, y )  (0, 0)

2
xe y
and P = (1,0,1). Find u such that Du [ f (1,0,1)] is a maximum.
2z2  3
Evaluate Du [ f (1,0,1)] for this vector u .
#7(6 pt’s) Let f ( x, y, z ) 
#8 (6 pts)
a. Find a function z  f ( x, y ) and a point (a, b) such that f (a, b)  (0, 0) . Be careful to clearly
state the point (a, b) . Please give a function f such that f ( x, y )  (0, 0) for all ( x, y ) .
b. What is true about your function (or the graph of your function) at the point (a, b) ? Explain.
c. Is your answer in b. a general result? That is, if f :
your answer in b. always be true?
2

with f (a, b)  (0, 0) , will
9. (3 pts) Find a function f :
3

such that
3 5 0
f ( p)  5 1 2  p,
0 2 6 
for all p 
3