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

912 Chapter 14 Partial Derivatives
effects of temperature and humidity. The heat index I is the perceived air temperature
when the actual temperature is T and the relative humidity is H. So I is a function of
T and H and we can write I − f sT, Hd. The following table of values of I is an excerpt
from a table compiled by the National Weather Service.
Table 1 Heat index I as a function of temperature and humidity
Relative humidity (%)
T H 50 55 60 65 70 75 80 85 90
90 96 98 100 103 106 109 112 115 119
Actual
temperature
(°F)
92
94
96
100
104
109
103
107
113
105
111
116
108
114
121
112
118
125
115
122
130
119
127
135
123
132
141
128
137
146
98
114
118
123
127
133
138
144
150
157
100
119
124
129
135
141
147
154
161
168
If 7et1403t01
we concentrate on the highlighted column of the table, which corresponds to a relative
04/28/10 humidity of H − 70%, we are considering the heat index as a function of the single
variable MasterID: T for a fixed 01568 value of H. Let’s write tsTd − f sT, 70d. Then tsTd de scribes how
the heat index I increases as the actual temperature T increases when the relative humidity
is 70%. The derivative of t when T − 968F is the rate of change of I with respect to
T when T − 968F:
t9s96d − lim
h l 0
ts96 1 hd 2 ts96d
h
− lim
h l 0
f s96 1 h, 70d 2 f s96, 70d
h
We can approximate t9s96d using the values in Table 1 by taking h − 2 and 22:
t9s96d <
ts98d 2 ts96d
2
−
f s98, 70d 2 f s96, 70d
2
−
133 2 125
2
− 4
t9s96d <
ts94d 2 ts96d
22
−
f s94, 70d 2 f s96, 70d
22
−
118 2 125
22
− 3.5
Averaging these values, we can say that the derivative t9s96d is approximately 3.75. This
means that, when the actual temperature is 968F and the relative humidity is 70%, the
apparent temperature (heat index) rises by about 3.758F for every degree that the actual
temperature rises!
Now let’s look at the highlighted row in Table 1, which corresponds to a fixed tempera
ture of T − 968F. The numbers in this row are values of the function GsHd − f s96, Hd,
which describes how the heat index increases as the relative humidity H increases when
the actual temperature is T − 968F. The derivative of this function when H − 70% is
the rate of change of I with respect to H when H − 70%:
G9s70d − lim
h l 0
Gs70 1 hd 2 Gs70d
h
− lim
h l 0
f s96, 70 1 hd 2 f s96, 70d
h
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