To find the number of distinct real roots of the cubic equation (1) x[caret]3 + bx[caret]2 + cx + d = 0, we could attempt to solve the equation. Fortunately, it is easy to tell the number of distinct real roots of (1) without having to solve the equation. The key is the discriminant. The discriminant of (1) appears in Cardan's (or Cardano's) cubic…

In a first calculus course, it is not unusual for students to encounter the theorems which state: If f is an even (odd) differentiable function, then its derivative is odd (even). In our paper, we prove some theorems which show how the symmetry of a continuous function f with respect to (i) the vertical line: x = a or (ii) with respect to the…

This article suggests the introduction of the concepts of areas bounded by plane curves and the volumes of solids of revolution in Pre-calculus. It builds on the basic knowledge that students bring to a pre-calculus class, derives a few more formulas, and gives examples of some problems on plane areas and the volumes of solids of revolution that…

To achieve the vision of mathematics set forth in "Crossroads" ("AMATYC," 1995), students must experience mathematics as a sensemaking endeavor that informs their world. Embedding the study of mathematics into the real world is a challenge, particularly because it was not the way that many of us learned mathematics in the first place. This article…

The orders of presentation of pre-calculus and calculus topics, and the notation used, deserve careful study as they affect clarity and ultimately students' level of understanding. We introduce an alternate approach to some of the topics included in this sequence. The suggested alternative is based on years of teaching in colleges within and…

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…

How might one define a functional operator D[superscript I]f(x), say for f(x) = 1 + x[superscript 2] + sin x, such that D[superscript +1](1 + x[superscript 2] + sin x) = 2x + cos x and D[superscript -1](1 + x[superscript 2] + sin x) = x + x[superscript 3]/3 - cos x? Our task in this article is to describe such an operator using a single formula…

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.…

Presents a dialog in question-and-answer form to explain the rationale behind using radians instead of degrees in calculus. (ASK)

The harmonic series is one of the most celebrated infinite series of mathematics. A quick glance at a variety of modern calculus textbooks reveals that there are two very popular proofs of the divergence of the harmonic series. In this article, the authors survey these popular proofs along with many other proofs that are equally simple and…

Describes the five-year evolution of a multi-sectioned precalculus course for business and health professions majors at the University of Hartford. Concludes that students have benefited from the revised course that uses the graphing calculator, calculator-based laboratory (CBL), and group work. (ASK)

We find infinite series in calculus to be one of the most confusing topics our students encounter. In this note, we look at some issues that our students find difficult or ambiguous involving the Ratio Test, the Root Test, and also the Alternating Series Test. We offer some suggestions and some examples, which could be a supplement to the set of…

Illustrates how Lagrange's approach applies to the differential calculus of polynomial functions when approximations are obtained. Discusses how to obtain polynomial approximations in other cases. (YP)

In a well-known calculus problem, an open top box is to be made from a rectangular piece of material by cutting equal squares from each corner and turning up the sides. The task is to find the dimensions of the box of maximum volume. Typically, the length of the sides of the corners that produces the largest volume turns out to be an irrational…