Engineering at Alberta Courses

Taylor Series: Applications

MacLaurin Series

The following are the MacLaurin series for some basic infinitely differentiable functions:

$\begin{split} e^x&=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots = \sum_{n=0}^\infty \frac{x^n}{n!}\\ \sin{x}&=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots = \sum_{n=0}^\infty \frac{(-1)^{n}x^{2n+1}}{(2n+1)!}\\ \cos{x}&=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots = \sum_{n=0}^\infty \frac{(-1)^{n}x^{2n}}{(2n)!} \end{split}$

Numerical Differentiation

Taylor’s Theorem provides a means for approximating the derivatives of a function. The first-order Taylor approximation of a function $f$ is given by:

$f(x)=f(a)+f'(a)h+\mathcal O (h^2)$

where $h=x-a$ is the step size. Then, :

$f'(a)=\frac{f(x)-f(a)}{h}+\frac{\mathcal O (h^2)}{h}=\frac{f(x)-f(a)}{h}+\mathcal O (h)$

Forward finite-difference

If we use the Taylor series approximation to estimate the value of the function $f$ at a point $x_{i+1}$ knowing the values at point $x_i<x_{i+1}$ then,we have:

(1) $\begin{equation*} f(x_{i+1})=f(x_{i})+f'(x_{i})h+\mathcal{O} (h^2) \end{equation*}$

where $h=x_{i+1}-x_i$ . In this case, the forward finite-difference can be used to approximate $f'(x_{i})$ as follows:

$f'(x_{i})=\frac{f(x_{i+1})-f(x_i)}{x_{i+1}-x_i}+\mathcal{O} (h)$

Backward finite-difference

If we use the Taylor series approximation to estimate the value of the function $f$ at a point $x_{i-1}$ knowing the values at point $x_i>x_{i-1}$ then, we have:

(2) $\begin{equation*} f(x_{i-1})=f(x_{i})-f'(x_{i})h+\mathcal O (h^2) \end{equation*}$

where $h=x_{i}-x_{i-1}$ . In this case, the backward finite-difference can be used to approximate $f'(x_{i})$ as follows:

$f'(x_{i})=\frac{f(x_{i-1})-f(x_i)}{x_{i-1}-x_i}+\mathcal O (h)$

Centred Finite Difference

The centred finite difference can provide a better estimate for the derivative of a function at a particular point. If the values of a function $f$ are known at the points $x_{i-1}<x_i<x_{i+1}$ and $x_{i+1}-x_i=x_i-x_{i-1}=h$ , then, we can use the Taylor series to find a good apprxoimation for the derivative as follows:

$\begin{split} f(x_{i+1})&=f(x_i)+f'(x_i) h+\frac{f''(x_i)}{2!}h^2+\mathcal{O}(h^3)\\ f(x_{i-1})&=f(x_i)+f'(x_i)(-h)+\frac{f''(x_i)}{2!}h^2+\mathcal{O}(h^3) \end{split}$

Subtracting the above two equations and dividing by $h$ gives the following:

$f'(x_i)=\frac{f(x_{i+1})-f(x_{i-1})}{2h}+\mathcal{O}(h^2)$

Building Differential Equations

See momentum balance and beam approximation for examples where the Taylor’s approximation is used to build a differential equation.

Approximating Continuous Functions

Taylor’s theorem essentially discusses approximating differentiable functions using polynomials. The approximation can be as close as needed by adding more polynomial terms and/or by ensuring that the step size $x-a$ is small enough. It is important to realize that polynomial approximations are valid for continuous functions that are not necessarily differentiable at every point. Let $f:[a,b]\rightarrow \mathbb{R}$ , then the Stone-Weierstrass Theorem states that for any small number $\varepsilon>0:\exists P(x)$ a polynomial function such that the biggest difference between $\max_{x\in[a,b]}|f(x)-P(x)|\leq \varepsilon$ . In other words, we can always find a polynomial function that approximates any continuous function on the interval $[a,b]$ within any degree of accuracy sought! This fact is extremely important in all engineering applications. Essentially, it allows modelling any continuous functions using polynomials. In other words, if a problem wishes to find the distribution of a variable (stress, strain, velocity, density, etc…) as a function of position, and if this variable is continuous, then, in the majority of applications we can assume differentiability and a Taylor approximation for the unknown function.