# Sample Exam #1 ```MVC
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Quiz #3
NAME:
1. Let f ( x, y)  ( x  2 y)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.
Evaluate each of the following limits, or show that it does not exist.
 x2 y 
a. Lim  4
 x , y    0,0 x  y 3 


b.
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Lim
 x , y    0,0
 x 2  y 2  Ln  x2  y 2 


3.
4.
 x2  4 y 2
if ( x, y)  (0,0)

Let G( x, y)   y ( x  2 y )
.
 0
if ( x, y )  (0,0)

a.
Determine with explanation if G is continuous at (0,0) .
b.
Find Gx (0,0) or show that it does not exist.
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.
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5.
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. 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.
6.
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Write a formula for
w
if w  f ( x, y, z ) and x  x(r , s ), y  y (r , s ), and z  z (r , s ) .
s
u
w
and y (u , v, w)  , then
v
v
z
w
z
u v
x
0
u
v
w
7. Prove that if z  f ( x, y ), with x(u, v, w) 
8. Let f ( x, y)  x  e2 y and P = (0,1,1). Find u such that Du [f(0,1,1)] is a maximum. Evaluate
Du [f(0,1,1)] for this vector u .
9. 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) .
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10.
The figure below shows the level curves of a function z  k  x, y  .
In this plot, the z-values increase as you move up and to the right.
a.
b.
On this contour plot, sketch vector v with initial point (2, 2) in the same direction as
k  2, 2  .
On this contour plot, sketch a vector w with initial point (2, 2) so that Dw k  2, 2  0 .
That is, the directional derivative at (2, 2) in the direction of w is 0.
k
k
c. For this contour plot, which has the greater value at the point (3, 1),
or
?
x
y