The Mean Value Theorem for Integrals is a powerful tool, which can be used to prove the Fundamental Theorem of Calculus, and to obtain the average value of a function on an interval. On the other hand, its weighted version is very useful for evaluating inequalities for definite integrals. This article shows the solutions on applying the weighted…

It is proved that if the differential equations "y[(n)] = f(x,y,y[prime],...,y[(n-1)])" and "y[(m)] = g(x,y,y[prime],...,y[(m-1)])" have the same particular solutions in a suitable region where "f" and "g" are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical…

The paper shows an alternative way of presenting differential calculus. It is shown that the Race Track Principles (RTP) (or any of the variants) is, in fact, equivalent to the Mean Value Theorem. Moreover, it is demonstrated how major theorems of differential calculus can be derived using RTP. The benefits of using RTP as a means to introduce…

The paper details a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts to force limit discussion into the language of individual real numbers and equality. The system of near-numbers…

Through numerical investigations, various examples of the Duffing type forced spring equation with epsilon positive, are studied. Since [epsilon] is positive, all solutions to the associated homogeneous equation are periodic and the same is true with the forcing applied. The damped equation exhibits steady state trajectories with the interesting…

First-year undergraduate engineering students' understanding of the units of factors and terms in first-order ordinary differential equations used in modelling contexts was investigated using diagnostic quiz questions. Few students appeared to realize that the units of each term in such equations must be the same, or if they did, nevertheless…

At the start of a freshman calculus course, many students conceive the classical definition of limit as the most problematic part of calculus. They not only find it difficult to understand, but also consider it of no use while solving most of the limit problems and therefore, skip it. This paper reformulates the rigorous definition of limit, which…

The general perception is that high school teaching of mathematics in South Africa tends to be fairly procedural and that students that enter university are better equipped to deal with procedural problems rather than conceptual. This study compares the conceptual and procedural skills of first-year calculus students in life sciences. Also…

This paper considers a second-year Mathematical Aspects in Architectural Design course, which relies on a first-year mathematics course and offers mathematical learning as part of hands-on practice in architecture design studio. The 16-hour course consisted of seminar presentations of mathematics concepts, their application to covering the plane…

This paper is a report on an attempt to teach students in their first and second year of university how to write mathematics. The problems faced by these students are outlined and the system devised to emphasize the importance of communicating mathematics is explained.

Offering mathematics majors the opportunity to engage in current, real-world applications can be an important enhancement to their undergraduate course curriculum. Instead of focusing on the traditional topic areas in pure and/or applied mathematics, one may structure a seminar course for senior mathematics majors by concentrating on a specific…

This note discusses the boundary in the frequency--amplitude plane for boundedness of solutions to the forced spring Duffing type equation. For fixed initial conditions and fixed parameter [epsilon] results are reported of a systematic numerical investigation on the global stability of solutions to the initial value problem as the parameters F and…

For many years, educators have been discussing benefits of educational practices such as the use of real-world examples, visualisation, interactivity, constructivism, self-paced learning and self-paced testing. Macromedia Flash MX has been used to develop online modules for the course Differential Equations offered at the University of South…