Although dogs seemingly follow the optimal path where they get to a ball thrown into the water, they certainly do not know the minimization function proposed in the calculus books. Trading the optimization problem for a related rates problem leads to a mathematically identical solution, which, it is argued here, is a more plausible model for the…

In his famous quadrature of the parabola, Archimedes found the area of the region bounded by a parabola and a chord. His method was to fill the region with infinitely many triangles each of whose area he could calculate. In his solution, he stated, without proof, three preliminary propositions about parabolas that were known in his time, but are…

Consider the series [image omitted] where the value of each a[subscript n] is determined by the flip of a coin: heads on the "n"th toss will mean that a[subscript n] =1 and tails that a[subscript n] = -1. Assuming that the coin is "fair," what is the probability that this "harmonic-like" series converges? After a moment's thought, many people…

This note considers functions of two variables which are continuous on a possibly unbounded closed region in [vertical bar]R[squared], and the functions of one variable obtained by integrating out the other variable over this region. The question of continuity of these functions is investigated, as are connections with joint density and marginal…

This note could find classroom use in an introductory course on complex analysis. Using some of the most significant theorems from complex analysis, the main result provides a simple method for transforming many elementary functions (defined on the complex plane) into everywhere continuous functions that are differentiable only on a nowhere dense…

In part 1 of this work we showed how modern mathematical research could, with a suitably chosen problem, be included in the first year curriculum of undergraduate mathematicians. With the use of Computer Algebra Systems, even the average undergraduate mathematician can aspire to discover interesting yet still unexplained behaviour in many areas of…

Recent developments are explained which use the concepts and methods of mathematical logic to provide a suitable framework for the development of the differential and integral calculus. (DT)

In terms of modern pedagogy, having visual interpretation of trigonometric functions is useful and quite helpful. This paper presents, pictorially, an easy approach to prove all single angle trigonometric identities on the axes. It also discusses the application of axial representation in calculus--finding the derivative of trigonometric functions.

Many students in U.S. schools have trouble understanding fractions and Algebra II, the one difficultly occurring at the end of elementary school, the other in high school. One reason is that schools generally focus on one aspect of mathematics--calculation--and often fail to address the second aspect--interpretation. Also responsible is the…

Popular culture provides many opportunities to develop quantitative reasoning. This article describes a junior-level, interdisciplinary, quantitative reasoning course that uses examples from movies, cartoons, television, magazine advertisements, and children's literature. Some benefits from and cautions to using popular culture to teach…

A Graeco-Latin square of order "n" is an "n[superscript x]n" array whose entries are the "n"[superscript 2] ordered pairs of numbers from 1 to "n", and in each row and each column the first elements of the ordered pairs are all different, as are the second elements. This article traces the history of the results that came out of work on a false…

Equivalence relations with a non-transparent transitive property are scarce in textbook examples. The equivalence relation given here is not only one with an interesting transitive property to prove, but also one that is instructive for students who are encountering proofs and relations for the first time.

A description of the development of three-valued logic includes construction of truth tables. (MK)

Resistance destroys symmetry. In this note, a graphical exploration serves as a guide to a rigorous elementary proof of a specific asymmetry in the trajectory of a point projectile in a medium offering linear resistance.

In the study reported here, we investigate the skills needed to validate a proof in real analysis, i.e., to determine whether a proof is valid. We first argue that when one is validating a proof, it is not sufficient to make certain that each statement in the argument is true. One must also check that there is good reason to believe that each…