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

930 Chapter 14 Partial Derivatives
In general, we know from (2) that an equation of the tangent plane to the graph of
a function f of two variables at the point sa, b, f sa, bdd is
z − f sa, bd 1 f x sa, bdsx 2 ad 1 f y sa, bdsy 2 bd
The linear function whose graph is this tangent plane, namely
3 Lsx, yd − f sa, bd 1 f x sa, bdsx 2 ad 1 f y sa, bdsy 2 bd
is called the linearization of f at sa, bd and the approximation
4 f sx, yd < f sa, bd 1 f x sa, bdsx 2 ad 1 f y sa, bdsy 2 bd
z
y
is called the linear approximation or the tangent plane approximation of f at sa, bd.
We have defined tangent planes for surfaces z − f sx, yd, where f has continuous first
partial derivatives. What happens if f x and f y are not continuous? Figure 4 pictures such
a function; its equation is
x
xy
f sx, yd x −H0
2 1 y 2
if
if
sx, yd ± s0, 0d
sx, yd − s0, 0d
FIGURE 44
f(x, f sx, y)= yd − xy xy
if (x,
if sx,
y)≠(0,
yd ±
0),
s0, 0d,
≈+¥ x 2 2
1 y
f(0, f s0, 0)=0 0d − 0
7et140404
05/03/10
MasterID: 01587
This is Equation 3.4.7.
You can verify (see Exercise 46) that its partial derivatives exist at the origin and, in fact,
f x s0, 0d − 0 and f y s0, 0d − 0, but f x and f y are not continuous. The linear approximation
would be f sx, yd < 0, but f sx, yd − 1 2 at all points on the line y − x. So a function of two
variables can behave badly even though both of its partial derivatives exist. To rule out
such behavior, we formulate the idea of a differentiable function of two variables.
Recall that for a function of one variable, y − f sxd, if x changes from a to a 1 Dx, we
defined the increment of y as
Dy − f sa 1 Dxd 2 f sad
In Chapter 3 we showed that if f is differentiable at a, then
5 Dy − f 9sad Dx 1 « Dx where « l 0 as Dx l 0
Now consider a function of two variables, z − f sx, yd, and suppose x changes from a
to a 1 Dx and y changes from b to b 1 Dy. Then the corresponding increment of z is
6 Dz − f sa 1 Dx, b 1 Dyd 2 f sa, bd
Thus the increment Dz represents the change in the value of f when sx, yd changes from
sa, bd to sa 1 Dx, b 1 Dyd. By analogy with (5) we define the differentiability of a function
of two variables as follows.
7 Definition If z − f sx, yd, then f is differentiable at sa, bd if Dz can be
expressed in the form
Dz − f x sa, bd Dx 1 f y sa, bd Dy 1 « 1 Dx 1 « 2 Dy
where « 1 and « 2 l 0 as sDx, Dyd l s0, 0d.
Definition 7 says that a differentiable function is one for which the linear approximation
(4) is a good approximation when sx, yd is near sa, bd. In other words, the tangent
plane approximates the graph of f well near the point of tangency.
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