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Taylor Polynomials

Level 1 - Math I (Physics) topic page in Power Series.

Principle

A Taylor polynomial is a finite polynomial chosen so that its value and derivatives match a function at one centre. It is the finite version of a Taylor series and is often the practical object used for approximation.

Taylor polynomial
\[P_N(x)=\sum_{n=0}^{N}\frac{f^{(n)}(c)}{n!}(x-c)^n\]

In physics, Taylor polynomials describe local behaviour: linear response, quadratic energy near equilibrium, and small-angle approximations are all Taylor-polynomial ideas.

Notation

\(P_N(x)\)
Taylor polynomial of degree N
\(c\)
centre of the approximation
\(f^{(n)}(c)\)
nth derivative of f at c
\(N\)
highest power kept
\(R_N(x)\)
remainder or error after degree N

The degree \(N\) is the highest power retained. Higher degree usually improves the approximation near the centre, but it also needs more derivative information.

Method

Step 1: Choose the centre and degree

Pick a centre \(c\) near the input values of interest. Choose \(N\) based on how many derivatives are available and how accurate the approximation needs to be.

Step 2: Compute derivatives at the centre

Find \(f(c), f'(c), f''(c),\ldots,f^{(N)}(c)\).

Step 3: Assemble the polynomial

Substitute the derivative values into the Taylor polynomial formula:

Degree 0 term
\[\frac{f(c)}{0!}(x-c)^0=f(c)\]
Degree 1 term
\[\frac{f'(c)}{1!}(x-c)=f'(c)(x-c)\]
Degree 2 term
\[\frac{f''(c)}{2!}(x-c)^2\]
Degree N term
\[\frac{f^{(N)}(c)}{N!}(x-c)^N\]

Rules

Degree N Taylor polynomial
\[P_N(x)=f(c)+f'(c)(x-c)+\frac{f''(c)}{2!}(x-c)^2+\cdots+\frac{f^{(N)}(c)}{N!}(x-c)^N\]
Maclaurin polynomial
\[c=0:\quad P_N(x)=\sum_{n=0}^{N}\frac{f^{(n)}(0)}{n!}x^n\]
Function split
\[f(x)=P_N(x)+R_N(x)\]

The polynomial is exact at \(x=c\) up to the derivative order used: \(P_N^{(k)}(c)=f^{(k)}(c)\) for \(0\le k\le N\).

Examples

Question
Find the quadratic Taylor polynomial for \(e^x\) about
\[0\]
Answer
All derivatives of \(e^x\) equal \(e^x\), so at
\[0\]
they equal
\[1\]
Therefore
\[P_2(x)=1+x+x^2/2\]

Checks

  • Taylor polynomials are finite sums.
  • The centre controls where the approximation is most accurate.
  • The coefficient of \((x-c)^n\) uses \(n!\) in the denominator.
  • A Taylor polynomial is not automatically a global approximation.

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Taylor Series