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

SectION 14.4 Tangent Planes and Linear Approximations 931
It’s sometimes hard to use Definition 7 directly to check the differentiability of a function,
but the next theorem provides a convenient sufficient condition for differentiability.
Theorem 8 is proved in Appendix F.
8 Theorem If the partial derivatives f x and f y exist near sa, bd and are continuous
at sa, bd, then f is differentiable at sa, bd.
Figure 5 shows the graphs of the
function f and its linearization L in
Example 2.
6
4
z
2
0
1
x
FIGURE 55
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0
1
0
y
_1
EXAMPLE 2 Show that f sx, yd − xe xy is differentiable at (1, 0) and find its linearization
there. Then use it to approximate f s1.1, 20.1d.
SOLUTION The partial derivatives are
f x sx, yd − e xy 1 xye xy
f y sx, yd − x 2 e xy
f x s1, 0d − 1 f y s1, 0d − 1
Both f x and f y are continuous functions, so f is differentiable by Theorem 8. The
lin earization is
Lsx, yd − f s1, 0d 1 f x s1, 0dsx 2 1d 1 f y s1, 0dsy 2 0d
− 1 1 1sx 2 1d 1 1 ? y − x 1 y
The corresponding linear approximation is
xe xy < x 1 y
so f s1.1, 20.1d < 1.1 2 0.1 − 1
Compare this with the actual value of f s1.1, 20.1d − 1.1e 20.11 < 0.98542.
EXAMPLE 3 At the beginning of Section 14.3 we discussed the heat index (perceived
temperature) I as a function of the actual temperature T and the relative humidity H
and gave the following table of values from the National Weather Service.
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
Find a linear approximation for the heat index I − f sT, Hd when T is near 968F and H
is near 70%. Use it to estimate the heat index when the temperature is 978F and the
relative humidity is 72%.
SOLUTION We read from the table that f s96, 70d − 125. In Section 14.3 we used the tabular
values to estimate that f T s96, 70d < 3.75 and f H s96, 70d < 0.9. (See pages 912–13.)
So the linear approximation is
f sT, Hd < f s96, 70d 1 f T s96, 70dsT 2 96d 1 f H s96, 70dsH 2 70d
< 125 1 3.75sT 2 96d 1 0.9sH 2 70d
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