01.07.1
Chapter 01.07
Taylor Theorem Revisited
After reading this chapter, you should be able to
1.
understand the basics of Taylor’s theorem,
2.
write transcendental and trigonometric functions as Taylor’s polynomial,
3.
use Taylor’s theorem to find the values of a function at any point, given the values of
the function and all its derivatives at a particular point,
4.
calculate errors and error bounds of approximating a function by Taylor series, and
5.
revisit the chapter whenever Taylor’s theorem is used to derive or explain numerical
methods for various mathematical procedures.
The use of Taylor series exists in so many aspects of numerical methods that it is imperative
to devote a separate chapter to its review and applications.
For example, you must have
come across expressions such as
+
−
+
−
=
!
6
!
4
!
2
1
)
cos(
6
4
2
x
x
x
x
(1)
+
−
+
−
=
!
7
!
5
!
3
)
sin(
7
5
3
x
x
x
x
x
(2)
+
+
+
+
=
!
3
!
2
1
3
2
x
x
x
e
x
(3)
All the above expressions are actually a special case of Taylor series called the Maclaurin
series.
Why are these applications of Taylor’s theorem important for numerical methods?
Expressions such as given in Equations (1), (2) and (3) give you a way to find the
approximate values of these functions by using the basic arithmetic operations of addition,
subtraction, division, and multiplication.
Example 1
Find the value of
25
.
0
e
using the first five terms of the Maclaurin series.
Solution
The first five terms of the Maclaurin series for
x
e
is
!
4
!
3
!
2
1
4
3
2
x
x
x
x
e
x
+
+
+
+
≈

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