Perhaps next time teachers head towards the fundamental theorem of calculus in their classroom, they may wish to consider Fermat's technique of finding expressions for areas under curves, beautifully outlined in Boyer's History of Mathematics. Pierre de Fermat (1601-1665) developed some important results in the journey toward the discovery of the…

The pre-equilibrium approximation is one of the several useful approximation methods that allows one to go from complex systems of differential equations to simple approximation methods. It is shown that the longtime behavior of systems subjected to pre-equilibrium can be obtained by a simple kinetic reasoning.

Two heuristic and three rigorous arguments are given for the fact that functions of the form Ce[kx], with C an arbitrary constant, are the only solutions of the equation dy/dx=ky where k is constant. Various of the proofs in this self-contained note could find classroom use in a first-year calculus course, an introductory course on differential…

A criterion (formula) for the termination of a continued fraction expansion leading to the solution of a standard differential eigenvalue problem from mathematical physics is presented. The criterion generates the eigenvalues in any specific case and is illustrated by elementary examples yielding well-known polynomial eigenfunctions. This…

This article describes some of the work of Jan Hudde who anticipated some results of calculus. Prior to a career as a Burgomaster of Amsterdam, Hudde engaged in mathematics. His method of finding maxima and minima is especially interesting.

Consider the family of monic polynomials of degree n having zeros at -1 and +1 and all their other real zeros in between these two values. This article explores the size of these polynomials using the supremum of the absolute value on [-1, 1], showing that scaled Chebyshev and Bernstein polynomials give the extremes.

This paper considers the problem of approximating an integrable function by piecewise linear functions that keep the integral and positivity of the original function.

In order to get undergraduates interested in mathematics, it is essential to involve them in its "discovery". In this paper, we will explain how technology and the knowledge of lower dimensional calculus can be used to help them develop intuition leading to their discovering the first derivative rule in multivariable calculus. (Contains 7 figures.)

Six proofs are given for the fact that for each integer n [greater than or equal to] 2, the nth root function, viewed as a function from the set of non-negative real numbers to itself, is not linear. If p is a prime number, then [Zeta]/p[Zeta] is characterized, up to isomorphism, as the only integral domain D of characteristic p such that D admits…

The focus of this paper is on student interpretation and usage of the existence and uniqueness theorems for first-order ordinary differential equations. The inherent structure of the theorems is made explicit by the introduction of a framework of layers concepts-conditions-connectives-conclusions, and we discuss the manners in which students'…

For various reasons there has been a recent trend in college and high school calculus courses to de-emphasize teaching the Partial Fraction Decomposition (PFD) as an integration technique. This is regrettable because the Partial Fraction Decomposition is considerably more than an integration technique. It is, in fact, a general purpose tool which…

PreCalculus students can use the Completing the Square Method to solve quadratic equations without the need to memorize the quadratic formula since this method naturally leads them to that formula. Calculus students, when studying integration, use various standard methods to compute integrals depending on the type of function to be integrated.…

Empirical research shows that students often use reasoning founded on copying algorithms or recalling facts (imitative reasoning) when solving mathematical tasks. Research also indicate that a focus on this type of reasoning might weaken the students' understanding of the underlying mathematical concepts. It is therefore important to study the…

This article discusses the assumptions required by the item response theory (IRT) true-score equating method (with Stocking & Lord, 1983; scaling approach), which is commonly used in the nonequivalent groups with an anchor data-collection design. More precisely, this article investigates the assumptions made at each step by the IRT approach to…

This article explores the use of problem posing in the calculus classroom using investigative projects. Specially, four examples of student work are examined, each one differing in originality of problem posed. By allowing students to explore actual questions that they have about calculus, coming from their own work or class discussion, or…