Edexcel IAL - Further Pure Mathematics 2- 6.3 Taylor Series- Study notes - New syllabus
Edexcel IAL – Further Pure Mathematics 2- 6.3 Taylor Series -Study notes- New syllabus
Edexcel IAL – Further Pure Mathematics 2- 6.3 Taylor Series -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 6.3 Taylor Series
Taylor Series
The Taylor series is a generalisation of the Maclaurin series. Instead of expanding a function about \( x=0 \), we expand it about any point \( x=a \).
If a function \( f(x) \) has derivatives of all orders at \( x=a \), then
\( f(x) = f(a) + f'(a)(x-a) + \dfrac{f”(a)}{2!}(x-a)^2 + \dfrac{f”'(a)}{3!}(x-a)^3 + \cdots \)
Why Taylor Series Are Useful
- To approximate functions near a chosen point.
- To simplify complicated expressions.
- To evaluate limits and integrals.
- To model physical systems.
Deriving the Expansion of \( \sin x \) about \( x=\pi \)
We want the Taylor series in powers of ( 𝑥 − 𝜋 ) (x−π).
Let \( f(x) = \sin x \). We calculate its derivatives at \( x=\pi \).
| \( f(x) \) | \( \sin x \) | \( f(\pi)=0 \) |
| \( f'(x) \) | \( \cos x \) | \( f'(\pi)=-1 \) |
| \( f”(x) \) | \( -\sin x \) | \( f”(\pi)=0 \) |
| \( f”'(x) \) | \( -\cos x \) | \( f”'(\pi)=1 \) |
Substitute into the Taylor formula:
\( \sin x = 0 – (x-\pi) + 0 + \dfrac{(x-\pi)^3}{3!} + \cdots \)
\( \sin x = -(x-\pi) + \dfrac{(x-\pi)^3}{6} + \cdots \)
Example
Find the Taylor series of \( e^x \) about \( x=1 \) up to \( (x-1)^2 \).
▶️ Answer / Explanation
\( e^x = e + e(x-1) + \dfrac{e}{2}(x-1)^2 \)
Example
Expand \( \cos x \) about \( x=0 \) up to \( x^4 \).
▶️ Answer / Explanation
\( \cos x = 1 – \dfrac{x^2}{2} + \dfrac{x^4}{24} \)
Example
Use the Taylor series of \( \sin x \) about \( x=\pi \) to approximate \( \sin 3.1 \).
▶️ Answer / Explanation
Let \( x = 3.1 \), \( x-\pi \approx -0.0416 \)
\( \sin x \approx -(x-\pi) + \dfrac{(x-\pi)^3}{6} \)
Substitute to get an approximate value.