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:

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

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

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:

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

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

In this course we will only need first-order Taylor approximations for linearizing non-linear functions of \(x\) being a vector with \(n\) variables. The corresponding Taylor polynomial 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 we need the \(n\) partial derivatives of function \(f\) evaluated at \(x_0\).