FP2 Chapter 6 – Maclaurin and Taylor Series
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INTRO :: Representing functions as polynomials
INTRO :: Representing functions as polynomials
Reasons why this is awesome
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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.
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Repeatedly differentiating functions
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Exercise 6A
Now onto the good stuff!
original func
power series
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Maclaurin Expansion
original func
power series
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Maclaurin Expansion
original func
power series
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(So the second derivatives matched already.)
Maclaurin Expansion
original func
power series
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Maclaurin Expansion
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These curves show the successive curves as we keep matching more and more derivatives.
Example
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Example
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Example
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Exercise 6B
Composite Functions
We can also apply these when the input to the function is different.
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Composite Functions
You might need some manipulation first.
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Composite Functions
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Exercise 6C
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
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Taylor Expansions
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Test Your Understanding
June 2009 Q5
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Exercise 6D
Solutions to differential equations
Bro Exam Note: This is by far the most common type of question in exams on this chapter.
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.
Solutions to differential equations
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Test Your Understanding
FP2 June 2012 Q5
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Exercise 6E
Summary
Official Edexcel specification