MVC
SHOW ALL WORK !!
Quiz #3
NAME:
y
1. Let f ( x, y )  e x
a. Find f x ( x, y ), f y ( x, y ), and f xy ( x, y )
b. Evaluate f (1, 0) .
c. Find the equation of the tangent plane to the graph of f at (1,0).
2.
Evaluate each of the following limits, or show that it does not exist.
 x2 y 
a. Lim  4
 x , y    0,0 x  y 2 


b.
MVC
Lim
 x , y    0,0 
 xy 


 x 2  y 2 
 x2  4 y 2
if ( x, y)  (0, 0)

Let G( x, y )   x  2 y
. Find Gx (0, 0) if it exists.
 1
if ( x, y)  (0, 0)

3.
4. Suppose that you have a function f :
( x, y )
(1, 2)
(1.02, 2)
(1,1.99)
2

with table of values given below:
f ( x, y )
4
4.1
4.3
a. Give an approximation for the tangent plane to the graph of f at (1,2).
b. Use the result from a. to approximate f (1.01, 2.01)
c. Under what conditions on f is the approximation from b. a “good” approximation? Explain.
MVC
Spherical/Cylindrical
Spherical/Cartesian

2
2
2
2
  x y z
 x   sin  cos 
 r   sin 

x2  y 2



y


sin

sin

tan








z
 z   cos 

 z   cos 


y

tan  

x
For each of the following, translate the given equation into the specified coordinate system, and
sketch its graph.
  2  r2  z2

r

 tan  
z





5.
  sin  cos  into Cartesian
6.
z  x 2  y 2 into both Cylindrical and Spherical
MVC
Spherical/Cylindrical
 r   sin 

  
 z   cos 

MVC
  2  r2  z2

r

 tan  
z





Spherical/Cartesian
 x   sin  cos 

 y   sin  sin 
 z   cos 


2
2
2
2
  x y z

x2  y 2

tan



z

y

tan  

x