For integrals consisting of rational functions of sine and cosine a set of little known rules known as the Bioche rules are considered. The rules, which consist of testing the differential form of the integral for invariance under one of three simple substitutions x [right arrow] -- x, [pi] -- x, and [pi] + x, allow one to decide which of the…

In this study, the conceptualizations used by prospective middle school mathematics teachers when defining, representing, and exemplifying the slope concept and relating it with other mathematical situations were examined. Participants' conceptualizations were identified by Nagle, C., Moore-Russo, D., Viglietti, J., & Martin, K. [(2013).…

This paper provides some insights into the use of example-generating tasks in the design of a technology-rich learning environment to enhance students' mathematical thinking. The paper reports on an early stage of a design-based research project concerning the design of tasks and associated feedback utilising the affordances provided by a combined…

Many students do not have a deep understanding of slope. This paper defines what a deep understanding of slope is in terms of mathematics-education theory. The various factors which help explain why such a deep understanding is difficult to acquire are then discussed. These factors include the following: the different representations for slope;…

Modern calculus textbooks carefully illustrate how to perform integration by trigonometric substitution. Unfortunately, most of these books do not adequately justify this powerful technique of integration. In this article, we present an accessible proof that establishes the validity of integration by trigonometric substitution. The proof offers…

Covariational reasoning has been the focus of many studies but only a few looked into this reasoning in the polar coordinate system. In fact, research on student's familiarity with polar coordinates and graphing in the polar coordinate system is scarce. This paper examines the challenges that students face when plotting polar curves using the…

In this paper, we report a pilot study on engaging a group of undergraduate students to explore the limits of sin(x)/x and tan(x)/x as x approaches to 0, with the use of non-graphic scientific calculators. By comparing the results in the pretest and the post-test, we found that the students had improvements in the tested items, which involved the…

In computing real-valued functions, it is ordinarily assumed that the input to the function is known, and it is the output that we need to approximate. In this work, we take the opposite approach: we show how to compute the values of some transcendental functions by approximating the input to these functions, and obtaining exact answers for their…

A closed form solution for the trigonometric integral [integral]sec[superscript 2k+1]xdx, k=0,1,2,..., is presented in this article. The result will fill the gap in another trigonometric integral [integral]sec[superscript 2m+1] x tan[superscript 2n]xdx, which is neglected by most of the calculus textbooks due to its foreseeable unorthodox solution…

This experiment was designed to determine which assessment method: continuous assessment (in the form of daily in-class quizzes), cumulative assessment (in the form of online homework), or project-based learning, best predicted student learning (dependent upon post-test grades) in an undergraduate mathematics course. Participants included 117…

We discuss a class of trigonometric functions whose corresponding Fourier series, on a conveniently chosen interval, can be used to calculate several numerical series. Particular cases are presented and two recent results involving numerical series are recovered. (Contains 1 note.)

We discuss a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series. Particular cases are presented. (Contains 4 notes.)

In the present note, two Parseval-type relations involving the Laplace transform are given. The application of the relations is demonstrated in evaluating improper integrals and Laplace transforms of trigonometric functions.

Using the divergence theorem technique of L. Eifler and N.H. Rhee, "The n-dimensional Pythagorean Theorem via the Divergence Theorem" (to appear: Amer. Math. Monthly), we extend the law of cosines for a triangle in a plane to an "n"-dimensional simplex in an "n"-dimensional space.

A complex technology-based problem in visualization and computation for students in calculus is presented. Strategies are shown for its solution and the opportunities for students to put together sequences of concepts and skills to build for success are highlighted. The problem itself involves placing an object under water in order to actually see…