Finite differences approximate derivatives using function values at discrete points. They are fundamental in numerical differentiation and solving differential equations.
Taylor's Theorem with Remainder:
For a smooth function $f(x)$ and point $x_0$, the Taylor expansion is:
$ f(x_0 + h) = f(x_0) + hf'(x_0) + \dfrac{h^2}{2!}f''(x_0) + \dfrac{h^3}{3!}f'''(x_0) + \cdots + \dfrac{h^n}{n!}f^{(n)}(x_0) + R_n(h) $
where the remainder term is $R_n(h) = \dfrac{h^{n+1}}{(n+1)!}f^{(n+1)}(\xi)$ for some $\xi \in (x_0, x_0 + h)$.
Common Finite Difference Formulas:
- Forward: $f'(x) \approx \dfrac{f(x+h) - f(x)}{h} + O(h)$
- Backward: $f'(x) \approx \dfrac{f(x) - f(x-h)}{h} + O(h)$
- Central: $f'(x) \approx \dfrac{f(x+h) - f(x-h)}{2h} + O(h^2)$ (higher accuracy!)
- Second derivative: $f''(x) \approx \dfrac{f(x+h) - 2f(x) + f(x-h)}{h^2} + O(h^2)$
Applications:
- Numerical solution of differential equations (finite difference methods)
- Computing derivatives from experimental or discrete data
- Image processing and edge detection
- Computational fluid dynamics and heat transfer simulations