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Finite Calculus (Calculus of Finite Differences): Definition, Example

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Finite calculus (also called calculus of finite differences) is an alternative to the “usual” differential calculus of infinitesimals that deals with discrete values. It’s called “finite” calculus because each is made up of a fixed (a.k.a. finite) set of terms.

Finite calculus is useful for many practical areas in science including:


  • Modeling and spreadsheets,
  • Marginal economic analysis,
  • Finance,
  • Growth and decay,
  • General computing.

It’s particularly useful for modeling human behavior and so is well-suited to many areas of social science.

Example of Step Size in Finite Calculus

One way to think of finite calculus is that it’s just calculus with infinity taken out of the picture. Instead of going all the way to the limit, finite calculus stops at a certain “step size” (Hamming, 2012). Instead of instantaneous, rates of change are discrete and finite (Morris & Stark, 2015).

The step size defines the difference between the two calculus branches. For example, instead of a unit change in x (Δx) with a step size approaching zero, the step is an entire unit of “x”. For example, from x to x + 1.

Example question: Find Δ f(x), if f(x) = x2.
Solution:
Our step size is x to x + 1, so we can find Δx by subtracting:
Δ f(x) = (x + 1)2 – x2 = 2x + 1.

History

“Finite Calculus” was the precursor to differential calculus and dates back to Brook Taylor (1717) and Jacob Stirling (1730) (Antosiewicz, 1977).


References

Antosiewicz, H. (1977). Studies in Ordinary Differential Equations, Volume 14. Mathematical Association of America.
Gleich, D. Finite Calculus: A Tutorial for Solving Nasty Sums. Retrieved October 27, 2019 from: https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20finite%20calculus.pdf
Hamming, R. (2012). Numerical Methods for Scientists and Engineers. Courier Corporation.
Morris, C. & Stark, R. (2015). Fundamentals of Calculus. John Wiley & Sons.
Watkins, T. The Summation of Series Using the Anti-Differencing Operation.

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Stephanie Glen. "Finite Calculus (Calculus of Finite Differences): Definition, Example" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/finite-calculus/
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