How to find the Taylor Polynomial of a function
The $n$-th degree Taylor Polynomial $T_n$ of a function $f(x)$ at a given point $x = a$ is defined as the polynomial that is closest to the function in an infinitesimally small neighbourhood of $x = a$.
There is a standard formula
$T_n(x) = f(a) + f'(a) (x-a) + \frac{f''(a)}{2!} (x-a)^2 +\cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n.$
In this HowDo, I will explain how to find the Taylor Polynomial in question using this formula. We will calculate the Taylor Polynomial of $f(x) = \sin{x}$ at $x = \frac{\pi}{3}$ to $3$rd degree as example.
What is known
Here I'd like to point out that the polynomial is determined not only by the function, but also by the degree $n$ of approximation and the center $x = a$.
| a function (sufficiently differentiable) | $f(x)$ |
| the center of approximation | $a$ |
| the degree of approximation | $n$ |
Step 1
Calculate derivatives up to $n$-th degree
For our example $f(x) = \sin{x}$, we make the following calculation by repeated differentiation.
| $f(x)$ | $\sin{x}$ |
| $f'(x)$ | $\cos{x}$ |
| $f''(x)$ | $-\sin{x}$ |
| $f'''(x)$ | $-\cos{x}$ |
Step 2
Evaluate the higher derivatives at $x=a$
For our example, $a = \frac{\pi}{3}$, so we have the following values
| $f(a)$ | $\frac{\sqrt{3}}{2}$ |
| $f'(a)$ | $\frac{1}{2}$ |
| $f''(a)$ | $-\frac{\sqrt{3}}{2}$ |
| $f'''(a)$ | $-\frac{1}{2}$ |
Step 3
Calculate the terms of $T_n$
Don't forget the factorials "!"
| $f(a)$ | $\frac{\sqrt{3}}{2}$ |
| $f'(a)(x-a)$ | $\frac{1}{2} (x-\frac{\pi}{3})$ |
| $\frac{f''(a)}{2!}(x-a)^2$ | $-\frac{\sqrt{3}}{4}(x-\frac{\pi}{3})^2$ |
| $\frac{f'''(a)}{3!}(x-a)^3$ | $-\frac{1}{12}(x-\frac{\pi}{3})^3$ |
Step 4
Add up all the terms
And we get this:
$T_3(x) = \frac{\sqrt{3}}{2} +\frac{1}{2} (x-\frac{\pi}{3}) -\frac{\sqrt{3}}{4}(x-\frac{\pi}{3})^2 -\frac{1}{12}(x-\frac{\pi}{3})^3.$
Taylor Polynomials of $\sin{x}$ at $x=0$
Taylor Polynomial vs. Taylor Series
One should carefully distinguish between Taylor Polynomial and Taylor Series. The first is a polynomial defined by the function $f$, a center of approximation $a$, and a degree of approximation $n$. Taylor Series, on the other hand, is a sum of terms of the form $\frac{f^{(n)}(a)}{n!}(x-a)^n$.
Taylor vs. MacLaurin
MacLaurin Polynomial refers to the particular case of $a = 0$. In other words, MacLaurin Polynomial is the Taylor Polynomial at $x = 0$.
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