5. Taylor series

5.1. Taylor’s theorem for approximating functions of 1 variable

Taylor’s theorem can be used to approximate a function \(f(x)\) with the so called \(p\)-th order Taylor polynomial:

\[ f(x) \approx f(x_0) +\frac{\partial}{\partial x}f(x_0)(x-x_0) + \frac{1}{2!} \frac{\partial^2}{\partial x^2}f(x_0)(x-x_0)^2 + \ldots +\frac{1}{p!} \frac{\partial^p}{\partial x^p}f(x_0)(x-x_0)^p=P_k (x) \]

where it is required that the function \(f: \mathbb{R}\mapsto \mathbb{R}\) is \(p\)-times differentiable at the point \(x_0 \in \mathbb{R}\).

The approximation error is equal to

\[ R_p(x) = f(x)- P_k (x) \]

and is called the remainder term.

Example:

A linear approximation (also called linearization) of \(f(x) = \cos(x)\) at \(x_0\) is obtained by the 1st order Taylor polynomial as:

\[ f(x) \approx \cos x_0 – \sin x_0 \cdot(x-x_0) \]

5.2. First-order Taylor polynomial for linearizing a function of \(n\) variables

For linearizing non-linear functions of \(x\) being a vector with \(n\) variables, we need the first-order Taylor polynomial, which is then given by:

\[ f(x) \approx f(x_0) + \partial_{x_1} f(x_0) \Delta x_0+ \partial_{x_2} f(x_0) \Delta x_0+ \ldots + \partial_{x_n} f(x_0) \Delta x_0 \]

where \(\Delta x_0=(x-x_0)\) and we need the \(n\) partial derivatives of function \(f\) evaluated at \(x_0\).