Taylor Polynomial: Formula & Examples

Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

In this lesson we explore the Taylor polynomial, which serves as a way to approximate a given function near a given point. Two examples will be worked out in detail.

What Is a Taylor Polynomial?

Let's start with the definition. Given a function f, a specific point x = a (called the center), and a positive integer n, the Taylor polynomial of f at a, of degree n, is the polynomial T of degree n that best fits the curve y = f(x) near the point a, in the sense that T and all its first n derivatives have the same value at x = a as f does.

If the center is 0, then T may be called a Maclaurin polynomial.

Graph of sin(x) (in blue) and its fifth degree Maclaurin polynomial (in red)
Graph of sin(x) and its fifth degree Maclaurin polynomial

Here is the formula to find the Taylor polynomial:

Formula for the Taylor Polynomial

Equivalently, if you expand the sum out, you get:

Expanded formula for Taylor polynomial

Now I know it may look daunting at first, but there's a step-by-step procedure for creating a Taylor polynomial. As long as you've had plenty of experience with derivatives, and if you know your way around factorials (that's the i! in the formula), then it shouldn't be too hard.

Computing a Taylor Polynomial

Notice that each term of the formula requires:

  1. A derivative of some order. In fact, the little (i) notation on the f means the derivative of order i (it doesn't mean f to the i power, even though the little (i) seems to look like an exponent).
  2. Plugging the given value a into that derivative.
  3. Dividing by the factorial number i!. Recall, i! = i(i - 1)(i - 2) .... (2)(1), if i > 0, and, by convention, 0! = 1.
  4. Multiplying by (x - a) to the i power. Note, in this formula x is just an unknown variable, and we leave it in the expression as part of the polynomial.

Once each term is built up, then the entire Taylor polynomial may be read off as the sum of those terms. Let's see this by example!

Example: Find the third degree Taylor polynomial for f(x) = 4/x, centered at x = 1.

First, we rewrite 4/x = 4x^(-1) to make derivatives easier to find. Notice the table below starts with i = 0. Always start with 0. In fact, it's the easiest part of the table, because this term simply boils down to f(a). Here, a = 1, because that's the given center.

i i-th deriv. Plug in center (a) Divide by i! Mult. by (x - a)^i
0 4x^-1 4(1)^-1 = 4 4/0! = 4/1 = 4 4
1 -4x^-2 -4(1)^-2 = -4 -4/1! = -4/1 = -4 -4(x - 1)
2 8x^-3 8(1)^-3 = 8 8/2! = 8/2 = 4 4(x - 1)^2
3 -24x^-4 -24(1)^-4 = -24 -24/3! = -24/6 = -4 -4(x - 1)^3

The answer is: T = 4 - 4(x - 1) + 4(x - 1)^2 - 4(x - 1)^3.

Another Example - The Natural Exponential

Probably one of the most important functions in calculus is the natural exponential, e^x, or as it is often written, exp(x). What makes this function so special is that the only function whose derivative is equal to itself is a constant multiple of exp(x). This fact makes finding Taylor polynomials of exp(x) quite easy!

Example: Find the fifth degree Maclaurin polynomial for exp(x).

Recall, a Maclaurin polynomial is simply a Taylor polynomial centered at a = 0.

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