Taylor's Formula
Taylor's Formula is one of the most powerful tools in calculus — it lets us approximate any smooth function using polynomials. Think of it as building a polynomial "clone" of a function that matches its behavior near a specific point.
🎯 The Big Idea
Imagine you're standing at a point $x = a$ on a function's graph. You know the function's value there, its slope, its concavity, and all its higher derivatives. Taylor's Formula uses all of this information to build a polynomial that approximates the function near that point.
The more derivative information you use (higher degree), the better the approximation becomes!
📐 Formal Definition
If $f(x)$ is a function that has derivatives of all orders at $x = a$, then the Taylor series of $f$ centered at $a$ is:
Written out explicitly:
The $n$-th degree Taylor polynomial $T_n(x)$ is the partial sum:
| Symbol | Meaning |
|---|---|
| $f^{(n)}(a)$ | The $n$-th derivative of $f$ evaluated at $x = a$ |
| $n!$ | $n$ factorial: $n! = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1$ |
| $a$ | The center of the expansion |
| $(x-a)^n$ | The displacement from center, raised to power $n$ |