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

948 Chapter 14 Partial Derivatives
EXAMPLE 1 Use the weather map in Figure 1 to estimate the value of the directional
derivative of the temperature function at Reno in the southeasterly direction.
SOLUTION The unit vector directed toward the southeast is u − si 2 jdys2 , but we
won’t need to use this expression. We start by drawing a line through Reno toward the
southeast (see Figure 4).
60
50
Reno
San
Francisco
60
70
70
Las
Vegas
80
FIGURE 4
0 50 100 150 200
(Distance in miles)
Los Angeles
7et140604 We approximate the directional derivative D u T by the average rate of change of the
temperature 05/04/10between the points where this line intersects the isothermals T − 50 and
TMasterID: − 60. The temperature 01604 at the point southeast of Reno is T − 608F and the temperature
at the point northwest of Reno is T − 508F. The distance between these points
looks to be about 75 miles. So the rate of change of the temperature in the southeasterly
direction is
D u T <
60 2 50
75
− 10
75 < 0.138Fymi ■
When we compute the directional derivative of a function defined by a formula, we
generally use the following theorem.
3 Theorem If f is a differentiable function of x and y, then f has a directional
derivative in the direction of any unit vector u − ka, bl and
D u f sx, yd − f x sx, yd a 1 f y sx, yd b
Proof If we define a function t of the single variable h by
tshd − f sx 0 1 ha, y 0 1 hbd
then, by the definition of a derivative, we have
tshd 2 ts0d
4 t9s0d − lim
− lim
h l 0 h
h l 0
f sx 0 1 ha, y 0 1 hbd 2 f sx 0 , y 0 d
h
− D u f sx 0 , y 0 d
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