12
DIFFERENTIATION
IN SEVERAL VARIABLES
12.1 Functions of Two or More Variables
Preliminary Questions
1.What is the difference between a horizontal trace and a level curve? How are they related?
2.Describe the trace off .x;y/Dx2sin.x3y/in thexz-plane.
3.Is it possible for two different level curves of a function to intersect? Explain.
4.Describe the contour map off .x; y/Dxwith contour interval 1.
5.
How will the contour maps of
f .x;y/
D
x
and
g.x;y/
D
2x
with contour interval 1 look different?
SOLUTION
The level curves of
f .x;y/
D
x
are the vertical lines
x
D
c
, and the level curves of
g.x; y/
D
2x
are the vertical
lines
2x
D
c
or
x
D
c
2
. Therefore, the contour map of
f .x; y/
D
x
with contour interval
1
consists of vertical lines with distance
one unit between adjacent lines, whereas in the contour map of
g.x; y/
D
2x
(with contour interval
1
) the distance between two
adjacent vertical lines is
1
2
.
Exercises
In Exercises 1–4, evaluate the function at the specified points.
1.
f .x;y/
D
x
C
yx
3
,
.2; 2/
,
.
1; 4/
SOLUTION
We substitute the values for
x
and
y
in
f .x;y/
and compute the values of
f
at the given points. This gives
f .2; 2/
D
2
C
2
2
3
D
18
f .
1; 4/
D
1
C
4
.
1/
3
D
5
2.
g.x;y/
D
y
x
2
C
y
2
,
.1; 3/
,
.3;
2/
SOLUTION
We substitute
.x; y/
D
.1; 3/
and
.x;y/
D
.3;
2/
in the function to obtain
g.1; 3/
D
3
1
2
C
3
2
D
3
10
I
g.3;
2/
D
2
3
2
C
.
2/
2
D
2
13
3.
h.x; y; z/
D
xyz
2
,
.3; 8; 2/
,
.3;
2;
6/
SOLUTION
Substituting
.x; y; z/
D
.3; 8; 2/
and
.x; y; z/
D
.3;
2;
6/
in the function, we obtain
h.3; 8; 2/
D
3
8
2
2
D
3
8
1
4
D
6
h.3;
2;
6/
D
3
.
2/
.
6/
2
D
6
1
36
D
1
6
1506
C H A P T E R
12
DIFFERENTIATION IN SEVERAL VARIABLES
4.
Q.y; z/
D
y
2
C
y
sin
z
,
.y; z/
D
2;
2
;
2;
6
SOLUTION
We have
Q
2;
2
D
2
2
C
2
sin
2
D
4
C
2
1
D
6
Q
2;
6
D
.
2/
2
2
sin
6
D
4
2
1
2
D
3
In Exercises 5–12, sketch the domain of the function.
