This paper illustrates a biological application of the concepts of relative change and area under a curve, from mathematics. We study two biological measures "relative change in cardiac output" and "cardiac output", which are predictors of heart blockages and other related ailments. Cardiac output refers to the quantity of…

This note offers a derivation of the Euler-Maclaurin formula that is simple and elementary. In addition, the paper shows that the derivation provides Euler-Maclaurin formulas for a variety of functionals other than the trapezoid rule.

When first-year calculus students are interested in studying double integrals, they can find, in standard textbooks, a detailed description of the different regions of integration. The aims of this paper are: to give a criterion to select the plane that will be projected, to classify the projections, and to give a simple rule to obtain them.…

The analysis of tautochrone problems involves the solution of integral equations. The paper shows how a reasonable assumption, based on experience with simple harmonic motion, allows one to greatly simplify such problems. Proposed solutions involve only mathematics available to students from first year calculus.

Three major techniques are employed to calculate [pi]. Namely, (i) the perimeter of polygons inscribed or circumscribed in a circle, (ii) calculus based methods using integral representations of inverse trigonometric functions, and (iii) modular identities derived from the transformation theory of elliptic integrals. This note presents a…

This article presents two classroom episodes in which students were exposed to the value of asking questions and to the different roles played by proof in mathematics. The conversation in the two episodes is outlined in the article. The setting was a classroom of fifteen good high-school students, who were studying calculus. These episodes…