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

SectION 14.4 Tangent Planes and Linear Approximations 929
TEC Visual 14.4 shows an animation
of Figures 2 and 3.
ing the domain of the function f sx, yd − 2x 2 1 y 2 . Notice that the more we zoom in, the
flatter the graph appears and the more it resembles its tangent plane.
40
40
40
20
z 0
_20
_4
_2
20
20
z 0
z 0
_20
_20
_2 _4 _2
0
0
_2
0
y 2
0
0
1
2 x
y
x
y
4 4
2 2
2 2
(a)
(b)
(c)
FIGURE 2 The elliptic paraboloid z − 2x 2 1 y 2 appears to coincide with its tangent plane as we zoom in toward s1, 1, 3d.
1
x
0
In Figure 3 we corroborate this impression by zooming in toward the point (1, 1) on a
contour map of the function f sx, yd − 2x 2 1 y 2 . Notice that the more we zoom in, the
more the level curves look like equally spaced parallel lines, which is characteristic of a
plane.
1.5
1.2
1.05
FIGURE 33
Zooming in toward (1, 1)
on a contour map of
f(x, f sx, y)=2≈+¥ yd − 2x 2 1 y 2 0.5
1.5
0.8
1.2
0.95
1.05
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MasterID: 01586
Linear Approximations
In Example 1 we found that an equation of the tangent plane to the graph of the function
f sx, yd − 2x 2 1 y 2 at the point s1, 1, 3d is z − 4x 1 2y 2 3. Therefore, in view of the
visual evidence in Figures 2 and 3, the linear function of two variables
Lsx, yd − 4x 1 2y 2 3
is a good approximation to f sx, yd when sx, yd is near s1, 1d. The function L is called the
linearization of f at s1, 1d and the approximation
f sx, yd < 4x 1 2y 2 3
is called the linear approximation or tangent plane approximation of f at s1, 1d.
For instance, at the point s1.1, 0.95d the linear approximation gives
f s1.1, 0.95d < 4s1.1d 1 2s0.95d 2 3 − 3.3
which is quite close to the true value of f s1.1, 0.95d − 2s1.1d 2 1 s0.95d 2 − 3.3225. But
if we take a point farther away from s1, 1d, such as s2, 3d, we no longer get a good
approxi mation. In fact, Ls2, 3d − 11 whereas f s2, 3d − 17.
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