Taylor's Theorem for One Variable Functions

The tangent line approximation is a first order approximation to a function. To get a higher order approximation by a polynomial we use Taylor's theorem. The approximation to f near the point (x₀,f(x₀)) is

$$f(x) \approx f(x_0) + \frac{f'(x_0)}{1!}(x - x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + \frac{f'''(x_0)}{3!}(x-x_0)^3 + \dots$$

The nth order term is easily seen to be (f⁽ⁿ⁾(x₀)/n!) (x-x₀)ⁿ.