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Section 2.8: Linear approximation and differentials
Main idea?
..?..
Derivatives are good for finding approximate values of
functions
If
f
is a differentiable function in the vicinity of
x
=
a
,
then: for
x
≈
a
, the (usually nonlinear) function
y
=
f
(
x
)
is well-approximated by the linear function
y
=
L
(
x
) =
f
(
a
)+
f
′
(
a
)(
x
–
a
),
aka the tangent line to the graph of
y
=
f
(
x
) at the point
(
a
,
f
(
a
)).
We call
L
(
x
) the linearization of
f
at
a
.
Equivalently,
f
(
a
+
h
) is well-approximated by
f
(
a
)+
f
′
(
a
)
h
for
h
approximately equal to 0.
Note that geometrically this is just saying that the graph of
the function is close to the tangent line, for points on the
graph near the point of tangency.
Example: For
x
100,
≈
y
=
f
(
x
)=sqrt(
x
) is close to
sqrt(100)+
C
(
x
–100) = 10+
C
(
x
–100),

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where
C
=
f
′
(100) = (1/2)(100)
–1/2
= 1/20.
E.g., sqrt(103) is close to 10+3/20=10.15.
Check this
mentally:
(10.15)
2
= 10
2
+ 2(10)(.15) + (.15)
2
= 100 + 3 + (.15)
2
103.
≈
This is often expressed notationally using differentials.

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