MVC
SHOW ALL WORK !!
Calculator Allowed.
Quiz #3
NAME:
1. Let f ( x, y)  ( xy  2)5
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.
3.
Evaluate the following limit or show that it does not exist.
 x2 y 
Lim
 x , y    0,0  x 4  y 3 


 x2  4 y 2
if ( x, y)  (0,0)

Let G ( x, y)   y ( x  2 y )
.
 0
if ( x, y )  (0,0)

Determine with explanation if G is continuous at (0,0) .
MVC
4.
Find the directional derivative of F ( x, y)  x 2  xy
in the direction of the vector
v  3i  4 j , at the point  1,3.
w
at the point (r , s)  (1, 1) if w  x 2 y  y 2 z and x  2r  s, y  r  2s, and z  2rs .
s
5.
Evaluate
6.
Find a tangent plane to the surface which is defined implicitly by the equation
xy 2  yz 2  zx 2  3 at the point (1, 2,1) .
MVC
For the contour map for z  f ( x, y ) shown below, estimate each of the following quantities.
Explain briefly how you are getting your answer.
(a) f x (1,2) and f y (1, 2)
(b) f (1, 2)
(c) Du  f (1,2) , where u is a unit vector in the direction of f (1, 2)
(d) Sketch the vector f (1, 2) on the contour map for f using (1,2) as the initial point.
(e) Sketch a unit vector v with initial point (1,2) such that Dv  f (1,2)  0 .
-1
0
1
2
3
4
5
MVC
Suppose that you have a function f :
7.
( 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).
c. Use the result from b. to approximate f (1.01, 2.01)
d. Under what conditions on f is the approximation from b. a “good” approximation?
Explain.
u
w
and y (u , v, w)  , then
v
v
z
z
z
u  v w
0
u
v
w
9. Prove that if z  f ( x, y ), with x(u, v, w) 
MVC
Concepts
8.
 2
 1 
2
if ( x, y )  (0,0)
 x  y  sin  2
2 
Let f ( x, y )  
x y 

0
if ( x, y )  (0,0)

a. Determine the partials of f at (0,0) , that is, f x (0,0) and f y (0,0) .
b. Are f x and f y continuous at (0,0)? You may discuss the continuity of only one of
these partials, as the other will be analogous. Explain/show work.
c. Determine whether f is differentiable at (0,0) . Again, Explain/show work.
MVC