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

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 was developed in the 17th century by Gottfried Wilhelm Leibniz and Isaac Newton. It was originally used to solve problems in physics and engineering, but it has since been applied to a wide variety of fields, including economics, finance, and computer science — where it is used to analyze the performance of algorithms..

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

• 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.

## Types of Finite Calculus

There are two main types of finite calculus: forward difference calculus and backward difference calculus. Forward difference calculus uses the differences between successive values of a function to approximate its derivative. Backward difference calculus uses the differences between the values of a function and its previous values to approximate its derivative.

## 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.

Finite calculus has a number of advantages over differential calculus. It is easier to understand and implement, and it is more accurate for discrete data. However, finite calculus is not as general as differential calculus, and it cannot be used to solve all problems.

## 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. Retrieved September 1, 2020 from: http://www.sjsu.edu/faculty/watkins/antidiff.htm

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