FP 2 Chapter 6 Maclaurin and Taylor Series

INTRO : : Representing functions as polynomials

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

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

Repeatedly differentiating functions ? ? ?

Now onto the good stuff! original func power series ?

Maclaurin Expansion original func power series ? ? ?

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

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

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

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

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

Test Your Understanding June 2009 Q 5 a? b?

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

Solutions to differential equations ? ? ?

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

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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 Reasons why this is awesome 1 2 3

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 ? ? ?

Exercise 6 A

Now onto the good stuff! original func power series ?

Maclaurin Expansion original func power series ? ? ?

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

Maclaurin Expansion original func power series ? ? ? ?

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

Example ? ? ? ? ?

Example ?

Exercise 6 B

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 ?

Exercise 6 C

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 ?

Test Your Understanding June 2009 Q 5 a? b?

Exercise 6 D

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 ? ? ?

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

Exercise 6 E

Summary Official Edexcel specification