10.11 Finding Taylor Polynomial Approximations of Functions

What is a Taylor Polynomial?

A Taylor polynomial approximates a function near a value x = a using derivatives at that point. The nth-degree Taylor polynomial is: P_n(x) = f(a) + f'(a)(x−a) + f''(a)/2!(x−a)² + ... + f^(n)(a)/n!(x−a)ⁿ. Higher degree usually means better approximation near a. When a = 0, it's called a Maclaurin polynomial.

Building the Polynomial Term by Term

Each term matches a derivative of f at x = a: the constant term matches f(a), the linear term matches f'(a), and so on. The coefficient of the (x−a)ⁿ term is f^(n)(a)/n!. This guarantees the polynomial and f share the same value and first n derivatives at x = a, making it a great local approximation.

Using Taylor Polynomials to Approximate Values

To estimate f(x) near a, plug x into P_n(x). Common examples include approximating e^x, sin(x), cos(x), or ln(x) using their Maclaurin polynomials. The more terms included, the smaller the typical error, especially closer to the center a. This is heavily tested alongside error bounds (Lagrange error).

Key terms

Taylor Polynomial (general form)

P_n(x) = f(a) + f'(a)(x−a) + f''(a)/2!(x−a)² + ... + f^(n)(a)/n!(x−a)ⁿ

Maclaurin Polynomial

A Taylor polynomial centered at x = 0.

Coefficient of (x−a)ⁿ term

f^(n)(a)/n! — the nth derivative at the center divided by n factorial.

Purpose of Taylor Polynomials

To approximate a function's value near a specific point using derivative information at that point.

3rd-degree Maclaurin polynomial for e^x

1 + x + x²/2 + x³/6

Try an AP-style question

What is the 2nd-degree Maclaurin polynomial for eˣ?

1. x + x²/2
2. 1 + x + x²
3. 1 + x²/2
4. 1 + x + x²/2