A1258690343StFMu

1
CHAPTER 6 TAYLOR POLYNOMIALS, TAYLOR SERIES AND
MACLAURIN SERIES
MRS EZRINDA MOHD ZAIHIDEE FAKULTI KEJ. ELEKTRIK
ELEKTRONIK
2
6.4 Taylor polynomials of the nth order
6.1 Introduction
6.2 Linearization using first order Taylor
polynomials
TAYLOR POLYNOMIALS, TAYLOR SERIES AND MACLAURIN
SERIES
6.5 Taylors formula and the remainder term
6.3 Second order Taylor polynomials
6.6 Taylor and Maclaurin series
2
3
6.4 Taylor polynomials of the nth order
6.1 Introduction
6.2 Linearization using first order Taylor
polynomials
TAYLOR POLYNOMIALS, TAYLOR SERIES AND MACLAURIN
SERIES
6.5 Taylors formula and the remainder term
6.3 Second order Taylor polynomials
6.6 Taylor and Maclaurin series
3
4
LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• explain the principle of superposition
• solve the given questions

4
5
6.1 INTRODUCTION

• Application Obtaining linearized models of
  non-linear systems ? much easier to analyze
• ? to make use of the

5
6
PRINCIPLE OF SUPERPOSITION
output
input
system
output
input
system
6
7
6.2 LINEARIZATION USING FIRST ORDER TAYLOR
POLYNOMIALS
y
f(x)
Q
x
a
0
7
8
6.2 LINEARIZATION USING FIRST ORDER TAYLOR
POLYNOMIALS
8
9
6.2 LINEARIZATION USING FIRST ORDER TAYLOR
POLYNOMIALS

• EXAMPLE
• A function, , and its first derivative are
  evaluated at
• State the first-order Taylor polynomial
  generated by at
• Estimate

9
10
6.2 LINEARIZATION USING FIRST ORDER TAYLOR
POLYNOMIALS
EXAMPLE Find a linear approximation to
near
10
11
6.2 LINEARIZATION USING FIRST ORDER TAYLOR
POLYNOMIALS
EXAMPLE Calculate the first-order Taylor
polynomial generated by about
11
12
6.2 LINEARIZATION USING FIRST ORDER TAYLOR
POLYNOMIALS
EXAMPLE Calculate the first-order Taylor
polynomial generated by about
Hence, evaluate and
12
13
6.4 Taylor polynomials of the nth order
6.1 Introduction
6.2 Linearization using first order Taylor
polynomials
TAYLOR POLYNOMIALS, TAYLOR SERIES AND MACLAURIN
SERIES
6.5 Taylors formula and the remainder term
6.3 Second order Taylor polynomials
6.6 Taylor and Maclaurin series
13
14
LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• explain the quadratic approximation
• solve the given questions

14
15
6.3 SECOND ORDER TAYLOR POLYNOMIALS
15
16
6.3 SECOND ORDER TAYLOR POLYNOMIALS
EXAMPLE Given estimate (a) (b)
16
17
6.3 SECOND ORDER TAYLOR POLYNOMIALS

• EXAMPLE
• (a) Obtain the second-order Taylor polynomial,
  
• generated by about
• Verify that
  and
• (c) Evaluate and

17
18
6.3 SECOND ORDER TAYLOR POLYNOMIALS
EXAMPLE A function, , is defined
by Obtain the second-order Taylor polynomial
generated by about
18
19
6.4 Taylor polynomials of the nth order
6.1 Introduction
6.2 Linearization using first order Taylor
polynomials
TAYLOR POLYNOMIALS, TAYLOR SERIES AND MACLAURIN
SERIES
6.5 Taylors formula and the remainder term
6.3 Second order Taylor polynomials
6.6 Taylor and Maclaurin series
19
20
LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• explain the pattern of the Taylors formula
• solve the given questions

20
21
6.4 TAYLOR POLYNOMIALS OF THE NTH ORDER
21
22
6.4 TAYLOR POLYNOMIALS OF THE NTH ORDER
EXAMPLE Given obtain a
fourth-order Taylor polynomial generated by
about Estimate
22
23
6.4 TAYLOR POLYNOMIALS OF THE NTH ORDER

• EXAMPLE
• A function, satisfies the equation
• Estimate using a third-order
  Taylor polynomial.
• (b) Estimate using a fourth-order
  Taylor polynomial.

23
24
6.4 TAYLOR POLYNOMIALS OF THE NTH ORDER
EXAMPLE Given obtain the third-,
fourth- and fifth-order Taylor polynomials
generated by about
24