LINEAR APPROXIMATION
Math21a, O. Knill
HOMEWORK: 13.7: 12, 16, 30, 34, 38
LINEAR APPROXIMATION.
1D:
The
linear approximation
of a function
f
(
x
) at a point
x
0
is the linear function
L
(
x
) =
f
(
x
0
) +
f
′
(
x
0
)(
x
−
x
0
)
.
The graph of
L
is tangent to the graph of
f
at
x
0
.
2D:
The
linear approximation
of a function
f
(
x, y
) at (
x
0
, y
0
)
is
L
(
x, y
) =
f
(
x
0
, y
0
) +
f
x
(
x
0
, y
0
)(
x
−
x
0
) +
f
y
(
x
0
, y
0
)(
y
−
y
0
)
The level curve of
g
is tangent to the level curve of
f
at (
x
0
, y
0
).
The graph of
L
is tangent to the graph of
f
.
3D:
The
linear approximation
of a function
f
(
x, y, z
) at
(
x
0
, y
0
, z
0
) by
L
(
x, y, z
)
=
f
(
x
0
, y
0
, z
0
) +
f
x
(
x
0
, y
0
, z
0
)(
x
−
x
0
) +
f
y
(
x
0
, y
0
, z
0
)(
y
−
y
0
) +
f
z
(
x
0
, y
0
, z
0
)(
z
−
z
0
)
.
The level surface of
L
is tangent to the level surface of
f
at
(
x
0
, y
0
, z
0
).
Using
∇
f
=
a
f
x
, f
y
A
, the linearization can be written as
L
(
vx
) =
f
(
vx
0
) +
∇
f
(
vx
0
)
·
(
vx
−
vx
0
)
HOW CAN IT BE USED? Linearization is important because linear functions are easier to deal with. Using
linearization, one can estimate function values near known points.
JUSTIFYING THE LINEAR APPROXIMATION.
If the second variable
y
=
y
0
is ±xed, then we have a onedimensional situation where the only variable is
x
. Now
f
(
x, y
0
) =
f
(
x
0
, y
0
) +
f
x
(
x
0
, y
0
)(
x
−
x
0
) is the linear approximation. Similarly, if
x
=
x
0
is ±xed
y
is the single
variable, then
f
(
x
0
, y
) =
f
(
x
0
, y
0
) +
f
y
(
x
0
, y
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This note was uploaded on 03/29/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.
 Spring '08
 Skrypka
 Calculus, Approximation, Linear Approximation

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