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18.01
Calculus
Jason
Starr
Fall
2005
Lecture
8.
September
27,
2005
Homework.
Problem
Set
2
all
of
Part
I
and
Part
II.
Practice
Problems.
Course
Reader:
2A1,
2A4,
2A9,
2A11,
2A12.
1.
Linear
approximations.
For
a
diﬀerentiable
function
f
(
x
),
the
linear
approximation
or
linearization
of
f
(
x
)
at
x
=
a
is
the
linear
function,
f
(
a
)
+
f
(
a
)(
x
−
a
)
.
In
a
precise
sense,
this
is
the
best
approximation
of
f
(
x
)
by
a
linear
function
near
x
=
a
.
For
x
close
to
a
,
the
value
of
f
(
x
)
is
close
to
the
value
of
the
linearization.
The
notation
for
this
is,
f
(
x
)
≈
f
(
a
)
+
f
(
a
)(
x
−
a
)
for
x
≈
a.
Example.
The
linearization
of,
f
(
x
)
=
e
−
3
x
sin(2
πx
)
+
5
e
−
3
x
cos(2
πx
)
,
near
x
=
0
is,
f
(
x
)
≈
5
−
(15
−
2
π
)
x
for
x
≈
0
.
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±
18.01
Calculus
Jason
Starr
Fall
2005
In
particular,
for
x
=
0
.
02,
this
gives
the
approximate
answer,
f
(0
.
02)
≈
5
−
(15
−
2
π
)(0
.
02)
≈
4
.
8.
The
actual
value
is
approximately
4
.
71.
2.
Basic
approximations.
Some
linear
approximations
occur
so
often,
they
should
be
committed
to
memory.
Each
of
the
following
is
the
linear
approximation
for
x
≈
0,
together
with
the
terms
in
the
quadratic
and
higher
approximations.
1
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This note was uploaded on 06/03/2008 for the course MATH B6A taught by Professor Moretti during the Spring '08 term at Bakersfield College.
 Spring '08
 Moretti
 Approximation, Linear Approximation, Mean Value Theorem

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