The most of the long distance courses are mainly based on two or three face to face meetings and printed courses or courses available on the online course web site. In this case the individual students study has attached an important amount of time. Usually the students do not receive a feed-back of their progress and do not receive answers to…

This article describes a hands-on approach to learning proofs by induction. Students create patterns of dominos so that when the first domino is pushed, the entire design collapses, one domino at a time. Students then build designs that do not work. By constructing a list of conditions that ensure a given pattern will collapse, students discover…

A linearization method of the nonlinear differential equation for conservative nonlinear oscillators is analysed and discussed. This scheme is based on the Chebyshev series expansion of the restoring force which allows us to obtain a frequency-amplitude relation which is valid not only for small but also for large amplitudes and, sometimes, for…

If students are in an advanced mathematics class, then at some point they enjoyed mathematics and looked forward to learning and practicing it. There is no reason that this passion and enjoyment should ever be lost because the subject becomes more difficult or rigorous. This author, who teaches advanced precalculus to high school juniors,…

We discuss a class of trigonometric functions whose corresponding Fourier series, on a conveniently chosen interval, can be used to calculate several numerical series. Particular cases are presented and two recent results involving numerical series are recovered. (Contains 1 note.)

This note presents two demonstrations of the known formula for the sum of squares of the first n natural numbers. One demonstration is based on geometrical considerations and the other one uses elementary integral calculus. Both demonstrations are very easy to understand, even for high school students, and may be good examples of how to explore…

We discuss a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series. Particular cases are presented. (Contains 4 notes.)

Using only fairly simple and elementary considerations--essentially from first year undergraduate mathematics--we show how the classical Stokes' theorem for any given surface and vector field in R[superscript 3] follows from an application of Gauss' divergence theorem to a suitable modification of the vector field in a tubular shell around the…

The standard approach to finding antiderivatives of trigonometric expressions such as sin(ax) cos(bx) is to make use of certain trigonometric identities. The disadvantage of this technique is that it gives no insight into the problem, but relies on students using a memorized formula. This note considers a technique for finding antiderivatives of…

Two original problems in geometry are presented with solutions utilizing to differential calculus: (a) rectangle inscribed in a sector; (b) point on the ray of the angle. The possibility of applying mathematics in general and differential calculus in particular for solution of practical problems is discussed. (Contains 8 figures.)

In this note, we give an alternative approach to introduce the mean value theorem. By first establishing Cauchy-type Mean Value Theorem, we can easily derive other Mean Value Theorem, and their consequences. We believe this approach is more effective in presenting this topic in a typical calculus class.

This paper addresses the forming of integral expressions such as [integral][superscript b][subscript a]f(x)dx by differential or infinitesimal derivations. A rigorous justification of these handy informal derivations is a major pedagogical problem in calculus. In this paper, checks are presented by which one can usually verify--with clear…

An acronym is presented that provides students a potentially useful, unifying view of the major topics covered in an elementary calculus sequence. The acronym (CAL) is based on viewing the calculus procedure for solving a calculus problem P* in three steps: (1) recognizing that the problem cannot be solved using simple (non-calculus) techniques;…

The basic rules of elementary algebra can often appear beyond the grasp of many students. Even though most subjects, including calculus, prove to be more difficult, it is the simple rules of algebra that continue to be the "thorn in the side" of many mathematics students. In this paper we present a result intended to help students achieve a…

A new class of Euler's method for the numerical solution of ordinary differential equations is presented in this article. The methods are iterative in nature and admit their geometric derivation from an exponentially fitted osculating straight line. They are single-step methods and do not require evaluation of any derivatives. The accuracy and…