March 2010 ContentsPutting Differentials Back Into Calculus
Tevian Dray, Corinne A. Manogue
We argue that the use of differentials in introductory calculus courses is useful and provides a unifying theme, leading to a coherent view of the calculus. Along the way, we meet several interpretations of differentials, some better than others.
Gröbner Basis Representations of Sudoku
Laura Taalman, Elizabeth Arnold, Stephen Lucas
This paper uses Gröbner bases to explore the inherent structure of Sudoku puzzles and boards. In particular, we develop three different ways of representing the constraints of Sudoku puzzles with a system of polynomial equations. In one case, we explicitly show how a Gröbner basis can be used to obtain a more meaningful representation of the constraints. Gröbner basis representations can be used to find puzzle solutions or count numbers of boards.
What’s My Domain?
This article gives a simple method for determining the maximum interval of existence for a solution of a single, autonomous, first-order differential equation as well as the behavior of the solution as the independent variable approaches the ends of the interval. The methods used are elementary enough to be included in an introductory differential equations course.
The Dance of the Foci
It’s well known that slicing a cone with a plane and then allowing the plane to rotate through all possible angles of inclination yields the conic sections. What paths then do the foci of these conics trace out as this cutting plane passes through the different angles? In this article, we derive formulae for these trajectories and generate the associated curves. Of particular interest is how they articulate when the conic shifts from ellipse to parabola to hyperbola.
The Locus of the Focus of a Rolling Parabola
Anurag Agarwal, James Marengo
The catenary is usually introduced as the shape assumed by a hanging ?exible cable. This is a physical description of a catenary. In this article we give a geometrical description of a catenary. Specifically we show that the catenary is the locus of the focus of a certain parabola as it rolls on the x-axis.
Four Ways to Skin a Definite Integral
Thomas Dence, Joseph Dence
The integral of 1/(1 + x2) is standard in elementary calculus, but the related integral 1/(1 + x4) rarely appears. In this article we examine the latter integral, computing its value by four different methods; several that involve standard elementary calculus techniques, and several involving complex integration.
POEMS in Newton’s Aerodynamic Frustum
Jaime Cruz Sampedro, Margarita Tetlalmatzi-Montiel
The golden mean is often naively seen as a sign of optimal beauty but rarely does it arise as the solution of a true optimization problem. In this article we present such a problem, demonstrating a close relationship between the golden mean and a special case of Newton’s aerodynamical problem for the frustum of a cone. Then, we exhibit a parallel relationship between the general case of this problem and a general family of means called POEM’s (for pth Order Extreme Means). The POEM’s share with the golden mean certain algebraic and geometric properties (established by Falbo), and also a remarkable minimizing physical property. Sadly, here is yet another situation in which an interesting property of the golden mean is not unique but is matched by the other means.
A Class of Multivariable Limits
Yingfan Liu, Youguo Wang
Students in multivariable calculus routinely are asked to evaluate limits of the form where mi and pi are positive integers for all i = 1, 2, . . ., n. In this note we prove a necessary and sufficient condition for the existence of all such limits.
Application of the Lambert W Function to the SIR Epidemic Model
The stable fixed point of the SIR epidemic model, or the asymptotic values of the population in susceptible, infected and removed classes, is expressed in terms of the Lambert W function. The epidemic threshold phenomenon is construed as a transcritical bifurcation.
PROBLEMS AND SOLUTIONS
Pythagoras Revenge by Arturo Sangalli, and The Housekeeper and the Professor by Yoko Ogawa
Reviewed by Susan Colley