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

Section 14.1 Functions of Several Variables 895
tions with the same temperature. Figure 13 shows a weather map of the world indicating
the average July temperatures. The isothermals are the curves that separate the colored
bands.
In weather maps of atmospheric pressure at a given time as a function of longitude
and latitude, the level curves are called isobars and join locations with the same pressure.
(See Exercise 34.) Surface winds tend to flow from areas of high pressure across the
isobars toward areas of low pressure, and are strongest where the isobars are tightly
packed.
A contour map of world-wide precipitation is shown in Figure 14. Here the level
curves are not labeled but they separate the colored regions and the amount of precipitation
in each region is indicated in the color key.
y
5
50
Example 9 A contour map for a function f is shown in Figure 15. Use it to estimate
the values of f s1, 3d and f s4, 5d.
4
3
2
1
80
70
60
50
80
70
60
SOLUTION The point (1, 3) lies partway between the level curves with z-values 70
and 80. We estimate that
f s1, 3d < 73
Similarly, we estimate that f s4, 5d < 56 ■
0
1 2 3 4 5
FIGURE 14 15
y
x
Example 10 Sketch the level curves of the function f sx, yd − 6 2 3x 2 2y for the
values k − 26, 0, 6, 12.
SOLUTION The level curves are
6 2 3x 2 2y − k or 3x 1 2y 1 sk 2 6d − 0
0
k=12
k=6
k=0
k=_6
x
This is a family of lines with slope 2 3 2 . The four particular level curves with
k − 26, 0, 6, and 12 are 3x 1 2y 2 12 − 0, 3x 1 2y 2 6 − 0, 3x 1 2y − 0, and
3x 1 2y 1 6 − 0. They are sketched in Figure 16. The level curves are equally spaced
parallel lines because the graph of f is a plane (see Figure 6).
FIGURE 15
Contour map of
f(x, y)=6-3x-2y
FIGURE 15 16
Contour map of of
f sx, f(x, yd y)=6-3x-2y
− 6 2 2 2y
7et140114-15
04/26/10
MasterID: 01539-40
y
0
k=12
k=0
k=6
Example 11 Sketch the level curves of the function
7et140114-15
04/26/10tsx, yd − s9 2 x 2 2 y 2 for k − 0, 1, 2, 3
MasterID: 01539-40
SOLUTION The level curves are
s9 2 x 2 2 y 2 − k or x 2 1 y 2 − 9 2 k 2
This is a family of concentric circles with center s0, 0d and radius s9 2 k 2 . The cases
k=_6
x
■
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