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

Section 14.1 Functions of Several Variables 889
x+y+1=0
y
above the line y − 2x 2 1, while x ± 1 means that the points on the line x − 1 must
be excluded from the domain. (See Figure 2.)
x=1
(b) f s3, 2d − 3 lns2 2 2 3d − 3 ln 1 − 0
_1
0
_1
x
Since lnsy 2 2 xd is defined only when y 2 2 x . 0, that is, x , y 2 , the domain of f is
D − hsx, yd | x , y 2 j. This is the set of points to the left of the parabola x − y 2 .
(See Figure 3.)
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FIGURE 2
œ„„„„„„„ x+y+1
Domain of f(x, f sx, y)= yd − sx 1 y 1 1
x-1 x 2 1
7et140102 y
04/23/10
MasterID: 01526
0
x=¥
x
Not all functions can be represented by explicit formulas. The function in the next
example is described verbally and by numerical estimates of its values.
Example 2 In regions with severe winter weather, the wind-chill index is often used
to describe the apparent severity of the cold. This index W is a subjective temperature
that depends on the actual temperature T and the wind speed v. So W is a function of
T and v, and we can write W − f sT, vd. Table 1 records values of W compiled by the
US National Weather Service and the Meteorological Service of Canada.
Table 1 Wind-chill index as a function of air temperature and wind speed
Wind speed (km/h)
FIGURE 33
Domain of f(x, f sx, y)=x yd − x ln(¥-x) lnsy 2 2 xd
7et140103
04/23/10
MasterID: The Wind-Chill 01527 Index
The wind-chill index measures how
cold it feels when it’s windy. It is based
on a model of how fast a human face
loses heat. It was developed through
clinical trials in which volunteers were
exposed to a variety of temper atures
and wind speeds in a refrigerated wind
tunnel.
Actual temperature (°C)
T v 5 10 15 20 25 30 40 50 60 70 80
5
0
5
10
15
20
25
30
35
40
4
2
7
13
19
24
30
36
41
47
3
3
9
15
21
27
33
39
45
51
2
4
11
17
23
29
35
41
48
54
1
5
12
18
24
30
37
43
49
56
1
6
12
19
25
32
38
44
51
57
0
6
13
20
26
33
39
46
52
59
1
7
14
21
27
34
41
48
54
61
1
8
15
22
29
35
42
49
56
63
2
9
16
23
30
36
43
50
57
64
2
9
16
23
30
37
44
51
58
65
3
10
17
24
31
38
45
52
60
67
For instance, the table shows that if the temperature is 258C and the wind speed is
50 kmyh, 7et1401t01 then subjectively it would feel as cold as a temperature of about 2158C with
no wind. 04/23/10 So
MasterID: 01528 f s25, 50d − 215
Example 3 In 1928 Charles Cobb and Paul Douglas published a study in which they
modeled the growth of the American economy during the period 1899–1922. They
considered a simplified view of the economy in which production output is determined
by the amount of labor involved and the amount of capital invested. While there are
many other factors affecting economic performance, their model proved to be remarkably
accurate. The function they used to model production was of the form
1 PsL, Kd − bL K 12
where P is the total production (the monetary value of all goods produced in a year),
L is the amount of labor (the total number of person-hours worked in a year), and K is
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