James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 14.1 Functions of Several Variables 897
K
300
200
Level curves are labeled with the value of the production P. For instance, the level
curve labeled 140 shows all values of the labor L and capital investment K that result
in a production of P − 140. We see that, for a fixed value of P, as L increases K
decreases, and vice versa.
■
100
100
100
FIGURE 18 19
220
180
140
200 300
L
For some purposes, a contour map is more useful than a graph. That is certainly
true in Example 13. (Compare Figure 19 with Figure 8.) It is also true in estimating function
values, as in Example 9.
Figure 20 shows some computer-generated level curves together with the corresponding
computer-generated graphs. Notice that the level curves in part (c) crowd
together near the origin. That corresponds to the fact that the graph in part (d) is very
steep near the origin.
y
z
z
7et140118
04/26/10
MasterID: 01543
x
x
y
(a) Level curves of f(x, y)=_xye _≈_¥
(b) Two views of f(x, y)=_xye _≈_¥
y
z
x
x
y
FIGURE 20
(c) Level curves of f(x, y)=
_3y
≈+¥+1
(d) f(x, y)=
_3y
≈+¥+1
Functions of Three or More Variables
A function of three variables, f , is a rule that assigns to each ordered triple sx, y, zd in a
domain D R 3 a unique real number denoted by f sx, y, zd. For instance, the temperature
T at a point on the surface of the earth depends on the longitude x and latitude y of
the point and on the time t, so we could write T − f sx, y, td.
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