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Approximation
Authors: Bill Davis, Horacio Porta and Jerry Uhl
©19962007
Publisher: Math Everywhere, Inc.
Version 6.0
3.04 Taylor's Formula
LITERACY
·
L.1)
All you know about a function f
@
x
D
is:
f
@
0
D
=
2, f
£
@
0
D
=
6, and f
££
@
0
D
= 
8.
Write down the expansion of f
@
x
D
in powers of x through the x
2
term.
·
L.2)
All you know about a function f
@
x
D
is:
f
@
2
D
=
1, f
£
@
2
D
= 
3, and f
££
@
2
D
=
1.
Write down the expansion of f
@
x
D
in powers of
H
x

2
L
through the
H
x

2
L
2
term.
·
L.3)
All you know about a pair of functions f
@
x
D
and g
@
x
D
is:
f
@
1
D
=
0, f
£
@
1
D
=
6, and g
@
1
D
=
0, and g
£
@
1
D
=
2 .
Calculate the lim
x
Ø
1
f
@
x
D
g
@
x
D
.
·
L.4)
All you know about a pair of functions f
@
x
D
and g
@
x
D
is:
f
@
1
D
=
0, f
£
@
1
D
=
0, f
££
@
1
D
=
8, g
@
1
D
=
0, g
£
@
1
D
=
0, and g
££
@
1
D
=
2.
Calculate lim
x
Ø
1
f
@
x
D
g
@
x
D
.
·
L.5)
Here is the expansion of
f
@
x
D
=
e
Sin
@
p
x
D
in powers of
x

1
through the
H
x

1
L
4
term:
Normal
A
Series
A
E
Sin
@
p
x
D
,
8
x, 1, 4
<EE
1
 p
H

1
+
x
L
+
1
2
p
2
H

1
+
x
L
2

1
8
p
4
H

1
+
x
L
4
Here is Taylor's formula for the expansion of a cleared function f
@
x
D
in powers of
H
x

1
L
through the
H
x

1
L
4
term:
Clear
@
f
D
;
Normal
@
Series
@
f
@
x
D
,
8
x, 1, 4
<DD
f
@
1
D
+
H

1
+
x
L
f
£
@
1
D
+
1
2
H

1
+
x
L
2
f
££
@
1
D
+
1
6
H

1
+
x
L
3
f
H
3
L
@
1
D
+
1
24
H

1
+
x
L
4
f
H
4
L
@
1
D
Use what you see above to write down the values of
f
£
@
1
D
,
f
££
@
1
D
,
f
H
3
L
@
1
D
, and
f
H
4
L
@
1
D
in the case that
f
@
x
D
=
e
Sin
@
p
x
D
.
·
L.6)
Here is the expansion of
f
@
x
D
=
Sin
@
Tan
@
x
DD

Tan
@
Sin
@
x
DD
in powers of x through the x
9
term:
Normal
@
Series
@
Sin
@
Tan
@
x
DD

Tan
@
Sin
@
x
DD
,
8
x, 0, 9
<DD

x
7
30

29 x
9
756
No, that's not a misprint.
How does this reveal that
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This note was uploaded on 10/11/2011 for the course MATH 241 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Kim
 Approximation

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