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I

.— - Session 2265

Integration of the Greatest Integer Function

David Fung and Steve Ligh

Department of Mathematics Southeastern Louisiana University Hammond LA 70402

There have been much discussions in recent years about the teaching of calculus. Among the various directions in the reform movement is the Consortium based at Harvard University. The text [2] for this project begins with a discussion of a library of functions. We introduced in our first semester calculus class several functions that are applicable to students’ environment. The postage stamp function and the grading function are examples of step functions, such as the greatest integer function.

The greatest integer function, oft, denoted by [t], is defined as [t] = n for every t c [n, n + 1) with n being an integer. Our study of the greatest integer function started with the use of the Computer Algebra System, Derive version 2.0. In order to study greatest integer function in Derive, one must first load the utility file MISC.MTH. The file contains the function FLOOR(a, b) which is defined as the greatest integer less than or equal to a/b. Therefore, the greatest integer function of t is given as FLOOR(t, 1). When we used the Derive command “Simplify” on this function, we obtained the following representation for [t]:

[t] = ~arctan(cot(7rt)) + t - ~. (1)

By means of elementary properties of trigonometric functions, we proved (1) for non-integral t. Representation (1) and its generalization have been studied by Fung and Ligh[l ].

Let F(z) = JO’ [t]dt. In this paper, we will consider F(z) and show how we can approximate it by two quadratic functions. In [1], by using (1) and the substitution u = ~arctan(cot(nt) ), we obtained the following integration formula:

- [x]’ + 2X[Z] - [x] F(x) o (2) L

The graph of F is given in Figure . Formula (2) was also found by Sy [3] using a different method.

, . -. -- . . - ...~&& .:4 } 1996 ASEE Annual Conference Proceedings ‘Q,.,pllll’;?

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Ligh, S., & Fung, D. (1996, June), Integration Of The Greatest Integer Function Paper presented at 1996 Annual Conference, Washington, District of Columbia. 10.18260/1-2--6134