Taylor Polynomials: An Introduction

What a Taylor Polynomial Is

Some functions are difficult to calculate exactly. For example, evaluating at a number such as is not as straightforward as evaluating a polynomial. Taylor polynomials give us a powerful way to approximate complicated functions using polynomials.

The basic idea is simple: choose a point , and construct a polynomial that matches the function's value, slope, curvature, and higher-order behavior at that point.

The closer you stay to the center , the more accurate the approximation generally becomes.

The Taylor Polynomial Formula

The Taylor polynomial of degree centered at is:

There is a pattern here. The first term uses the function itself, the second uses the first derivative, the third uses the second derivative, and so on. Each derivative is evaluated at the center .

Why the Formula Works

Suppose you want a polynomial that approximates a function near . Start with a constant that matches the function's value:

But matching the value alone does not guarantee that the polynomial has the same slope. So we add a linear term involving :

To also match the second derivative, we add a quadratic term. The factorial appears because differentiating a power repeatedly produces factors that need to be canceled:

Continuing this pattern gives the complete Taylor polynomial.

A Step-by-Step Method

1. Identify the function , the center , and the requested degree .
2. Find the derivatives through .
3. Evaluate every derivative at the center .
4. Substitute into the Taylor formula.
5. Simplify the polynomial and use it to approximate the function near .

Worked Examples

Taylor Polynomial vs. Taylor Series

A Taylor polynomial stops after a finite number of terms. For example, is a polynomial of degree 4.

A Taylor series continues indefinitely:

So the Taylor polynomial is a finite approximation, while the Taylor series is the infinite expansion that can represent the function within its interval of convergence.

Maclaurin Polynomials

A Maclaurin polynomial is simply a Taylor polynomial centered at . Because all the derivatives are evaluated at zero, the formula becomes especially convenient:

Many of the most important AP Calculus BC series come from Maclaurin series, including those for , , and .

How Degree Affects the Approximation

A higher-degree Taylor polynomial generally matches more of the function's local behavior. A degree-1 polynomial matches the function's value and slope. A degree-2 polynomial also matches curvature. A degree-3 polynomial matches the next level of behavior, and so on.

For a point sufficiently close to the center, adding more terms can make the approximation dramatically more accurate.

Common Mistakes

Practice Problems