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MATH1010:
Chapter 3 cont… and Chapter 4
1
DIFFERENTIATION RULES cont…
Linear Approximations and Differentials (3.11, pg. 262) cont…
Recall:
Last day, we introduced the equation for finding a linear approximation.
Example:
Find a linearization of
x
x
f
1
)
(
=
at
2
=
a
.
Notes:
•
what we’re doing is making sure that the values of
)
(
x
f
and
)
(
x
L
at the point
a
x
=
match, as do the values of the slopes there
•
)
(
x
L
is a function of
x
.
•
The approximation is better the closer
x
is to
a
.

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MATH1010:
Chapter 3 cont… and Chapter 4
2
•
Higher order approximations are possible.
Question:
How does the behaviour of the approximation compare to that of the original
function?
Recall:
dx
dy
x
f
=
′
)
(
We call
dx
and
dy
differentials
, and they are related through
dx
x
f
dy
)
(
′
=
Geometrically:
One can think of
dy
as the amount that the tangent line rises or falls when
x
changes by
dx
, while
y
Δ
is the amount that the curve rises or falls when
x
changes by
dx.

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