In the multivariable calculus course, a standard application of triple integration is to find volumes of bounded regions in R[superscript 3]. In this article, we consider the problem of computing volumes, by way of examples, of regions bounded by planes and quadric surfaces. For illustration, we present the solution to two basic but non-standard…

Much of the education research on implicit differentiation and related rates treats the topic of differentiating equations as an unproblematic application of the chain rule. This paper instead problematizes the legitimacy of this procedure. It develops a conceptual analysis aimed at exploring how a student might come to understand when and why one…

When Johann Bernoulli published his lectures on integrals in 1742, integral calculus had become very advanced since the time of their composition in 1692. Nevertheless, these lectures are of excellent clarity and simplicity even when the book deals with major problems of Mathematical Physics. Just to pique some interest, we offer a commented…

Following a modeling-first approach to differential equations, inquiry-based learning activities on modeling and controlling the growth of locally relevant species, such as lionfish or sea turtles, are developed by the authors. Emerging exemplary practices are presented, building towards a balanced teaching of differential equations involving…

Recently a teacher friend enquired about the S-I-R equations for disease spread, and what follows was stimulated by that exchange. COVID-19 provides an opportunity to put mathematical flesh on verbal bones such as "self-isolation", "lockdown", "herd immunity", "flattening the curve", "closed…

In this article, the authors begin by considering symbolic literacy in mathematics. Next, they examine the origins of misuse of the equals sign by primary and junior secondary students, where "=" has taken on an operational meaning. They explain that in algebra, students need both the operational and relational meanings of the equals…

We estimate the parameters present in several differential equation models of population growth, specifically logistic growth models and two-species competition models. We discuss student-evolved strategies and offer "Mathematica" code for a gradient search approach. We use historical (1930s) data from microbial studies of the Russian biologist,…

A continuous representation of a hybrid method with three "off-step" points is developed via interpolation and collocation procedures, and used to obtain initial value methods (IVMs) for solving initial value problems. The IVMs are assembled into a single block matrix equation which is convergent and A-stable. We note that accuracy is improved by…

More often than one might at first imagine, a simple trick involving integration by parts can be used to compute indefinite integrals in unexpected and amusing ways. A systematic look at the trick illuminates the question of whether the trick is useful enough to be called an actual technique of integration.

In this article a continuous-time stochastic model (the Ornstein-Uhlenbeck process) is presented to model the perpetually altering states of the core affect, which is a 2-dimensional concept underlying all our affective experiences. The process model that we propose can account for the temporal changes in core affect on the latent level. The key…

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…

Recent drug studies have investigated the incidence of blood clots among patients using the Ortho Evra[R] birth control patch. In this article, the author describes an investigation of oestrogen levels in the body resulting from the application of the Ortho Evra[R] birth control patch versus daily use of Ortho-Cyclen[R] birth control pills.…

The solutions of the discrete logistic growth model based on a difference equation and the continuous logistic growth model based on a differential equation are compared and contrasted. The investigation is conducted using a dynamic interactive spreadsheet. (Contains 5 figures.)

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…

As the title says, this article considers the dog-on-the-beach problem from the perspective of the calculus of variations, making connections with the brachistochrone problem and Snell's law.