FP 2 Chapter 6 Maclaurin and Taylor Series

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FP 2 Chapter 6 – Maclaurin and Taylor Series Dr J Frost (jfrost@tiffin. kingston.

FP 2 Chapter 6 – Maclaurin and Taylor Series Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com Last modified: 5 th February 2016

INTRO : : Representing functions as polynomials

INTRO : : Representing functions as polynomials

INTRO : : Representing functions as polynomials Reasons why this is awesome 1 2

INTRO : : Representing functions as polynomials Reasons why this is awesome 1 2 3

Repeatedly differentiating functions We’ll be able to generate these power series using something called

Repeatedly differentiating functions We’ll be able to generate these power series using something called a Taylor expansion. This requires us to keep differentiating a function (to get each extra term of the polynomial). Let’s practice this first. ? ?

Repeatedly differentiating functions ? ? ?

Repeatedly differentiating functions ? ? ?

Exercise 6 A

Exercise 6 A

Now onto the good stuff! original func power series ?

Now onto the good stuff! original func power series ?

Maclaurin Expansion original func power series ? ? ?

Maclaurin Expansion original func power series ? ? ?

Maclaurin Expansion original func power series ? ? ? (So the second derivatives matched

Maclaurin Expansion original func power series ? ? ? (So the second derivatives matched already. )

Maclaurin Expansion original func power series ? ? ? ?

Maclaurin Expansion original func power series ? ? ? ?

Maclaurin Expansion 1 5 3 9 7 These curves show the successive curves as

Maclaurin Expansion 1 5 3 9 7 These curves show the successive curves as we keep matching more and more derivatives.

Example ? ? ? ? ?

Example ? ? ? ? ?

Example ? ? ? ? ?

Example ? ? ? ? ?

Example ?

Example ?

Exercise 6 B

Exercise 6 B

Composite Functions We can also apply these when the input to the function is

Composite Functions We can also apply these when the input to the function is different. ?

Composite Functions You might need some manipulation first. Expansion ? Valid Interval ?

Composite Functions You might need some manipulation first. Expansion ? Valid Interval ?

Composite Functions ?

Composite Functions ?

Exercise 6 C

Exercise 6 C

Taylor Expansions Bro Exam Note: There hasn’t been a single exam question to date

Taylor Expansions Bro Exam Note: There hasn’t been a single exam question to date which is based purely on the stuff you’ve done up to now in this chapter. From now on you’re very much in exam territory!

Taylor Expansions

Taylor Expansions

Taylor Expansions ? ? ?

Taylor Expansions ? ? ?

Taylor Expansions ?

Taylor Expansions ?

Test Your Understanding June 2009 Q 5 a? b?

Test Your Understanding June 2009 Q 5 a? b?

Exercise 6 D

Exercise 6 D

Solutions to differential equations Just as some functions can’t be integrated (in terms of

Solutions to differential equations Just as some functions can’t be integrated (in terms of elementary mathematical functions), similarly many differential equations can’t be integrated. But we can use a Taylor series to approximate the solution. Bro Exam Note: This is by far the most common type of question in exams on this chapter.

Solutions to differential equations ? ? ?

Solutions to differential equations ? ? ?

Test Your Understanding FP 2 June 2012 Q 5 a? b?

Test Your Understanding FP 2 June 2012 Q 5 a? b?

Exercise 6 E

Exercise 6 E

Summary Official Edexcel specification

Summary Official Edexcel specification

TAYLOR SERIES Maclaurin series center is 0 TaylorTAYLOR SERIES Maclaurin series center is 0 Taylor
Taylor and Maclaurin series is called the TaylorTaylor and Maclaurin series is called the Taylor
Maclaurin and Taylor Series Power Series Objective ToMaclaurin and Taylor Series Power Series Objective To
Taylor and Maclaurin Series Convergent Power Series FormTaylor and Maclaurin Series Convergent Power Series Form
Chapter 2 Power Taylor and Maclaurin Series LecturesChapter 2 Power Taylor and Maclaurin Series Lectures
Chapter 2 Power Taylor and Maclaurin Series LecturesChapter 2 Power Taylor and Maclaurin Series Lectures
Taylor CHAPTER and Maclaurin 2 Series Theorem IfTaylor CHAPTER and Maclaurin 2 Series Theorem If
Taylor and Maclaurin Series Lesson 9 10 ConvergentTaylor and Maclaurin Series Lesson 9 10 Convergent
Maclaurin and Taylor Series Introduction A wide rangeMaclaurin and Taylor Series Introduction A wide range
NUMERICAL ANALYSIS Maclaurin and Taylor Series Preliminary ResultsNUMERICAL ANALYSIS Maclaurin and Taylor Series Preliminary Results
Maclaurin and Taylor Series Introduction A wide rangeMaclaurin and Taylor Series Introduction A wide range
Section 9 7 Infinite Series Maclaurin and TaylorSection 9 7 Infinite Series Maclaurin and Taylor
Section 9 7 Infinite Series Maclaurin and TaylorSection 9 7 Infinite Series Maclaurin and Taylor
Section 8 68 7 Taylor and Maclaurin SeriesSection 8 68 7 Taylor and Maclaurin Series
11 10 Taylor and Maclaurin Series The purpose11 10 Taylor and Maclaurin Series The purpose
Maclaurin Taylor Series KUS objectives BAT Find theMaclaurin Taylor Series KUS objectives BAT Find the
Lesson 74 Introduction to Taylor Maclaurin Series HLLesson 74 Introduction to Taylor Maclaurin Series HL
TAYLOR MACLAURIN SERIES 9 10 MARCH 15 THTAYLOR MACLAURIN SERIES 9 10 MARCH 15 TH
TAYLOR AND MACLAURIN q how to represent certainTAYLOR AND MACLAURIN q how to represent certain
MACLAURIN AND TAYLOR POLYNOMIALS DAY 1 Section 9MACLAURIN AND TAYLOR POLYNOMIALS DAY 1 Section 9
Taylor Series Revisited 1 Taylor Series Revisited WhatTaylor Series Revisited 1 Taylor Series Revisited What
Serie de Taylor y Maclaurin Ing Antonio CrivilleroSerie de Taylor y Maclaurin Ing Antonio Crivillero
BROOK TAYLOR 1685 1731 BROOK TAYLOR Brook TaylorBROOK TAYLOR 1685 1731 BROOK TAYLOR Brook Taylor
Frederick Taylor Paula Benevene Taylor Frederick Taylor FiladelfiaFrederick Taylor Paula Benevene Taylor Frederick Taylor Filadelfia