Finite Difference Calculus

Finite difference calculus provides a discrete analog to continuous derivatives. It operates on sequences and discrete functions (such as integer polynomials) rather than continuous functions. Forward differences measure the change between adjacent terms, allowing polynomials to be reconstructed perfectly using discrete binomial expansions instead of continuous Taylor series.

Experiment: Build Exact Difference Tables

Open the Finite Difference Calculus module to explore the differences of integer polynomials.

1. Select Polynomial degree 3, Integer n 2, and Difference order 1. Read the Forward difference as 19 and the Continuous derivative as 12. Observe the highlighted node in the difference triangle visualization.
2. Change the Difference order to 3. Notice the Forward difference is now 6. For f(n) = n³, the third difference is a constant 3! = 6.
3. Set the Difference order to 5. The Forward difference becomes 0, because differences of order greater than the polynomial degree vanish perfectly in discrete calculus.
4. Change the Polynomial degree to 4 and Difference order to 2 (keeping n = 2). The Forward difference is 110, while the Continuous derivative is 48.
5. Change the Polynomial degree to 0, Integer n to 0, and Difference order to 1. Notice the polynomial evaluates to the constant 1 even at n = 0, and the forward difference is 0.