(10/7/08)
Math 10C. Lecture Examples.
Section 14.3. Local linearity and the differential
†
Zooming in on level curves of a nonlinear
z
=
f
(
x
,
y
)
If a function
y
=
f
(
x
) of one variable has a derivative at
x
0
and the graph
y
=
f
(
x
) is generated by a
calculator or computer in a small enough window containing the point (
x
0
, f
(
x
0
)), the displayed portion
of the graph will look like a line. This occurs because the graph is closely approximated by the tangent
line near that point.
Graphs of functions of two variables with continuous first derivatives are closely approximated by
planes in small windows. Consequently, their level curves at equal
z
increments look like equally spaced
parallel lines in small windows. This is illustrated by the level curves of
K
(
x, y
) = 3
x
2
y
3
+
x
in Figures 1
through 3. The level curves look more like equally spaced parallel lines in Figure 2 than Figure 1, and
even more like equally spaced parallel lines in Figure 3. These approximate closely the level lines of the
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 Spring '07
 Hohnhold
 Calculus, Derivative, maximum possible error

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