Taylor Polynomials

Taylor Polynomials Overview

Sometimes to avoid working with very complex functions, people use approximations. The most commonly used approximation is the linear or first order approximation (see the article linear approximations and differentials, for more information about this).

Let's look at a quick example.

So, how can you get a better approximation that works for values farther away?

Well, you can think of the following: the linear approximation works because you know the rate of change at a point, if you knew how the rate of change varies (i.e., the second derivative), that might provide you more information about what the function is like.

In general, if you knew how “all” the derivatives of the function behave, you could know exactly what the function is like. That is the idea behind Taylor polynomials.

Taylor Polynomials Formula

Let's state the definition of the Taylor polynomial.

There is a special case when \(x=0\) because it is much easier to write.

It is worth mentioning that the Taylor polynomials allow you to approximate any function using powers of \(x\). How good an approximation is it? That will depend on how many derivatives you calculate, the more derivatives, the more accurate the approximation.

Second Degree Taylor Polynomial

A particular case is when you want to approximate a function using the second derivative.

To see how a quadratic approximation is better than a linear approximation, let's approximate the function \(f(x)=\sin x\) at \(x=\dfrac{\pi}{2}\). First, let's calculate the derivatives and evaluate the functions at \(x=\dfrac{\pi}{2}\).

Table 1. Function and derivative values for \(\sin x\) at \(\dfrac{\pi}{2}\).

Thus, the first degree Taylor polynomial for \(f(x)=\sin x\) at \(x=\dfrac{\pi}{2}\) is

\[T_1(x)=1,\]

and the second degree Taylor polynomial for \(f(x)=\sin x\) at \(x=\dfrac{\pi}{2}\) is

\[T_2(x)=1-\frac{(x-\frac{\pi}{2})^2}{2!}.\]

Figure 2. Graph of the function \(f(x)=\sin x\) with its first and second degree Taylor polynomial.

Note that approximations also allow you to estimate values of a function at points where it is difficult to evaluate. For example, you know that \(\sin(\frac{\pi}{2})\) is equal to \(1\), but what about the value of \(\sin 2\)? Using the quadratic approximation, you get

\[\begin{align} \sin 2 &\approx T_2(2) \\ &=1-\frac{(2-\frac{\pi}{2})^2}{2!} \\ &\approx 0.9. \end{align}\]

❗❗ Remember that a Taylor polynomial centered at \(x=a\), only allows you to estimate values near \(x=a\).

Third Degree Taylor Polynomial

You could also approximate a function using the third derivative.

Let's calculate the third degree Taylor polynomial for the function \(g(x)=\sqrt{x}\) at \(x=1\).

Table 2. Derivatives and function values for \(g(x) = \sqrt{x}\).

Thus, its third degree Taylor polynomial centered at \(x=1\) is

\[T_3(x)=1+\frac{(x-1)}{2}-\frac{(x-1)^2}{8}+\frac{3(x-1)^3}{48}.\]

Figure 3. Graph of the function \(g(x)=\sqrt{x}\) with its Taylor polynomial of third degree.

Taylor Polynomials Examples

Let's look at more examples of how to use Taylor polynomials to estimate values of a function.

Let's look at another example.