A simple mathematical model for how vehicles follow each other along a stretch of road is presented. The resulting linear second-order differential equation with constant coefficients is solved and interpreted. The model can be used as an application of solution techniques taught at first-year undergraduate level and as a motivator to encourage…

The transition ('gap') between secondary and tertiary education in mathematics is a complex phenomenon covering a vast array of problems and issues. The aim of this paper is to present the ways in which the issues of mathematics transition have been dealt with at McMaster University. Roughly, the process of transition has been broken into three…

A series defining the antiderivative of an n th order differentiable function is defined. This series provides an explicit expression for the second part of the Fundamental Theorem of Calculus and can facilitate the establishment of new antiderivative functions.

This article discusses the mean value theorem and its importance to calculus. It focuses on Flett's mean value theorem. It includes five theorems with proofs and two examples.

An engineering problem concerned with relating the coordinates of the centre of a spherical cutting tool to the actual cutting surface leads to a potentially rich example of problem-solving techniques. Basic calculus, Lagrange multipliers and vector calculus techniques are employed to produce solutions that may be compared to better understand…

The aim of this note is to give some insight into the formal unity of a very applicable area of mathematics by showing an interesting analogy between the weak part of the Rouche-Frobenius theorem and the existence result for the initial value problem for the general first-order linear two-dimensional PDE.

Mathematical software packages such as Polymath, MATLAB, and Mathcad are currently widely used for engineering problem solving. Applications of several of these packages to typical chemical engineering problems have been demonstrated by Cutlip, et al. The main characteristic of these packages is that they provide a "problem-solving environment…

Calculus is one of those topics in mathematics where the algorithmic manipulation of symbols is easier than understanding the underlying concepts. Around 1680 Leibniz invented a symbol system for calculus that codifies and simplifies the essential elements of reasoning. The calculus of Leibniz brings within the reach of an ordinary student…

This study was inspired by a history of student difficulties in calculus, and innovation in response to those difficulties. The goals of the study were fourfold. First, to design a mathlet for students to explore local linearity. Second, to redesign the curriculum of first semester calculus around the use of technology, an emphasis on…

The purpose of this study was to investigate the effects of motivation, volition and belief change strategies, implemented with personal and group email messages, on students' attitudes, study habits and achievement in a calculus course for non-mathematics majors. Eighty four undergraduates enrolled in a calculus course received emails over a…

This paper introduces a concept of differentiability for functions in several variables which has a very natural definition and shares all the properties one expects of a notion of differentiability, but is shown to be weaker than the usual concept of differentiability defined as local linearizability. The exact connection between the two…

This paper explores the development of the method of partial fraction decomposition from elementary number theory through calculus to its abstraction in modern algebra. This unusual perspective makes the topic accessible and relevant to readers from high school through seasoned calculus instructors.

In the usual treatment of waves in introductory courses, one begins with traveling waves and the frequency/wavelength relationship f[lambda] = v, where "v" is the wave velocity. One then makes the point about superposition and shows that two waves traveling in opposite directions can add up to a standing wave; Eq. (1) still applies. This approach…

Several years ago I had to select a new textbook for my calculus-based introductory physics class. I subscribe to Just-in-Time Teaching methods,1 which require students to read the book before the material is covered in class. Thus, the readability of the text by the students is critical. However, I did not feel that I was the best judge of this…

This article presents a project for a differential equations class using an inverse problem based on the second order differential equation that models an electrical circuit. (Contains 2 footnotes.)