In the literature of mean and covariance structure analysis, noncentral chi-square distribution is commonly used to describe the behavior of the likelihood ratio (LR) statistic under alternative hypothesis. Due to the inaccessibility of the rather technical literature for the distribution of the LR statistic, it is widely believed that the…

We designed and implemented curriculum intended to be used by students in an algebra-based introductory physics laboratory course. Our curricular goal was to foster, through observations in the lab, a coherent framework in students' understanding of general principles presented in the introductory mechanics course, while addressing known student…

Currently, 92% of community colleges offer courses online, yet only 13% offer online mathematics developmental courses. Little research has been conducted to verify the effectiveness of such courses. This study attempted to answer: Are online developmental mathematics courses addressing the academic requirements (retention, success, and…

MATLAB is a powerful package for numerical computation. MATLAB contains a rich pool of mathematical functions and provides flexible plotting functions for illustrating mathematical solutions. The course of calculus-based business mathematics consists of two major topics: 1) derivative and its applications in business; and 2) integration and its…

Each year, well over a million students take college algebra and related courses. Very few of these students take the courses to prepare for calculus, but rather because they are required by other disciplines or to fulfill Gen Ed requirements. The present article discusses what the current mathematical needs are in most of those disciplines,…

I discuss my experience with teaching an advanced undergraduate Real Analysis class using both lecturing and the small-group guided discovery method. The article is structured as follows. The first section is about the organizational and administrative components of the class. In the second section I give examples of successes and difficulties…

Efforts to reform and improve teaching and learning in the undergraduate science, technology, engineering, and mathematics (STEM) disciplines have grown increasingly stronger and more focused over the past two decades. Since the early 1990s, some notable unifying developments have given coherence to such initiatives, as well as other developments…

The mean value theorem for differentials is basically a consequence of the fundamental theorem of calculus for Lebesgue integration.

Using data from the Longitudinal Study of American Youth (LSAY), we examined the extent to which students' mathematics coursework regulates (influences) the rate of growth in mathematics achievement during middle and high school. Graphical analysis showed that students who started middle school with higher achievement took individual mathematics…

Planimeters are devices that measure the area enclosed by a curve, and they come in a variety of forms. In this article, three of these, the rolling, polar, and radial planimeters, are described, and Green's theorem is used to show why they work.

Imagine trying to paint a picture with three colors--say red, blue, and yellow--with a blue region between any red and yellow regions, a red region between any blue and yellow regions, and a yellow region between any red and blue regions, down to infinitely fine details. Regions arranged in this way satisfy what is called the Wada property. At…

This article considers an old version of the classical pursuit problem as posed by Francis Godwin in 1599, who imagines a wedge of swans flying from the earth to the moon in twelve days, always flying at constant speed toward the moon. The return trip, during which the swans always fly toward the earth at the same speed, takes eight days. A little…

Since most students "hate" the concept of limit, in order to make them "happier," this article suggests a couple of naive "lim (h[right arrow]0)-is-missing" problems for them to try for fun. Indeed, differential functional equations that are related to difference quotients in calculus are studied in this paper. In particular, two interesting…

Calculus is a mathematical concept that is fundamental to how we understand the world around us. Whether it is in the world of technology, finance, astronomy, sociology, medicine, calculus in one form or another can be found. This brief article describes the origins of calculus in Greece, further developments by Newton and Leibniz, and the…

The following note contains the following sections: Introduction; The Kronecker Product and Some Formulae; and Proof of Formula. It discusses in detail the Kronecker product and how it relates to differential systems.