This paper deals with a brief historical introduction to fractional calculus with basic ideas, definitions and results of fractional order integration and differentiation of arbitrary order.

When first-year calculus students are interested in studying double integrals, they can find, in standard textbooks, a detailed description of the different regions of integration. The aims of this paper are: to give a criterion to select the plane that will be projected, to classify the projections, and to give a simple rule to obtain them.…

The Bernoulli polynomials are generalized and some properties of the resulting generalizations are presented.

This article describes several ways that calculus students can develop a deeper understanding of related rates by getting actively involved in the lesson.

The theory and application of second-order linear ordinary differential equations is reviewed from the standpoint of the operator factorization approach to the solution of ordinary differential equations (ODE). Using the operator factorization approach, the general second-order linear ODE is solved, exactly, in quadratures and the resulting…

This paper reports studies of the interaction between the intuitive, the formal and the procedural aspects in the processes of mathematical understanding of Peter, a first-year undergraduate of Mathematics. Using an activity and an interview, an attempt is made to analyse his mental operations. The way in which he handles visual-graphic…

We present an approach of defining certain transcendental functions as solutions to initial value problems or systems of such problems. This material is suitable for use in a second-semester one-variable calculus course.

How might one define a functional operator D[superscript I]f(x), say for f(x) = 1 + x[superscript 2] + sin x, such that D[superscript +1](1 + x[superscript 2] + sin x) = 2x + cos x and D[superscript -1](1 + x[superscript 2] + sin x) = x + x[superscript 3]/3 - cos x? Our task in this article is to describe such an operator using a single formula…

The appendix on the centres of gravity that appears at the end of Galileo's book, "Two New Sciences", is analysed. It is shown that the method used by Galileo in this work has an interesting reasoning and also shows preliminary ideas about scaling and advances some ideas about series convergence. In addition, we note that the geometrical language…

This article presents a way of computing a surface integral when the vector field of the integrand is a curl field. Presented in some advanced calculus textbooks such as [1], the technique, as the author experienced, is simple and applicable. The computation is based on Stokes' theorem in 3-space calculus, and thus provides not only a means to…

We investigate the possibility of approximating the value of a definite integral by approximating the integrand rather than using numerical methods to approximate the value of the definite integral. Particular cases considered include examples where the integral is improper, such as an elliptic integral. (Contains 4 tables and 2 figures.)

In this article, we discuss an alternative method of teaching students about the Dirac [delta]-function. The method provides students with the mechanical tools they need in order to work with the [delta]-function in practice, while also fostering a sense of cohesion in the calculus curriculum by presenting the [delta]-function as an evolution of…

This article shows how highly mobile computing, when used with new forms of network connectivity, can allow new forms of activities in the mathematics classroom. Examples are provided, such as the ability to share, harvest, and aggregate mathematical objects, and the ability for teachers and students to analyze the entire set of classroom…

Since antiquity, ideas regarding true beauty have been usurped by the purview of mathematics. From the aesthetic "logic" of Aristotle to the instrumentalized brutality of the Final Solution and its methodical anthropometric measurements, we see how the symmetry of numbers has been used to quantify the bodily politic according to an empirical…

Calculus-based business mathematics is a required quantitative course for undergraduate business students in most AACSB accredited schools or colleges of business. Many business students, however, have relatively weak mathematical background or even display math-phobia when presented with calculus problems. Because of the popularity of Excel, its…