September 2011 Contents
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In the September issue of The College Mathematics Journal, Stephen M. Walk takes the Intermediate Value Theorem apart and puts it back together again in The Intermediate Value Theorem is NOT Obvious—and I Am Going to Prove It to You. Also Mark Schwartz dishes out the latest on the mathematical prowess of dogs in Do Dogs Know the Trammel of Archimedes. Among other items of note, Greg N. Frederickson folds polyominoes, Annalisa Crannell analyzes the Shad-Fack transom window, and Dan Joseph, Gregory Hartman and Caleb Gibson introduce generalized parabolas. The issue also includes two Flaws, Fallacies, and Flimflam items and a Student Research Project on idealized coinage systems by Jack E. Graver.
The Intermediate Value Theorem Is NOT Obvious—and I Am Going to Prove It to You
Stephen M. Walk
The Intermediate Value Theorem is not often proved in Calculus I classes because many teachers and students see the theorem as obvious and its proof as impenetrable. This article addresses those two misconceptions, showing howthe IVT canbe proved in Calc I . . . and whyit shouldbe.
An Empirical Approach to the St. Petersburg Paradox
Dominic Klyve and Anna Lauren
The St. Petersburg game is a probabilistic thought experiment. It describes a game which seems to have infinite expected value, but which no reasonable person could be expected to pay much to play. Previous empirical work has centered around trying to find the most likely payoff that would result from playing the game ntimes. In this paper, we extend this work to the distribution of all possible values which could result from this experiment. We use this distribution—with a surprising fractal-like pattern—to examine the unlikely nature of the most famous experiment on this game, the results of the Compte de Buffon’s playing the game 2048 times.
Folding Polyominoes from One Level to Two
Greg N. Frederickson
For any given polyomino, is it possible to cut it into pieces and then hinge the pieces, so that the polyomino folds up into a similar version of itself but two levels thick? While we don’t know how to do this for every polyomino, the article does show how to cut, hinge, and fold polyominoes from several infinite classes, providing an inexhaustible supply of puzzles that are both fun to solve and beautiful to fold.
Dan Joseph, Gregory Hartman, and Caleb Gibson
In this article we explore the consequences of modifying the common definition of a parabola by considering the locus of all points equidistant from a focus and (not necessarily linear) directrix. The resulting derived curves, which we call generalized parabolas, are often quite beautiful and possess many interesting properties. We show that the conic sections are generalized parabolas whose directrix is a circle or a line, and their well known reflective properties are actually specific instances of a reflective property held by all generalized parabolas. Finally, we offer the reader suggestions for further investigation.
Series with Inverse Function Terms
Finding the sum of a series in the form of a closed expression has always been a challenging problem in analysis. The paper presents an elementary method for summation of series with terms generated by functions satisfying subtraction identities.
From the Dance of the Foci to a Strophoid
The intersection of a plane and a cone is a conic section and rotating the plane leads to a family of conics. What happens to the foci of these conics as the plane rotates? A classical result gives the locus of the foci as an oblique strophoid when the plane rotates about a tangent to the cone. The analogous curve when the plane intersects a cylinder, in which case all the sections are ellipses, is a right strophoid. This article discusses both results and provides elementary geometric proofs. Rotation about a different axis, such as one meeting the axis of the cone or cylinder, gives a very different curve. We consider how the resulting curve relates to the classical one by analyzing the family of curves obtained as the axis of rotation moves.
Do Dogs Know the Trammel of Archimedes?
The refraction problem, well-known in calculus and physics, continues to reveal new insights. This paper presents a geometric solution in which the trammel of Archimedes plays the prominent role. When properly configured, the trammel generates an ellipse and its family of normal lines. One normal line in particular intersects the boundary separating the media, the intersection point being the solution. The trammel, implemented as a rolling circle mechanism, can be constructed physically and/or programmed as a computer animation.
The Shad-Fack Transom
We provide several constructions, both algebraic and geometric, for determining the ratio of the radii of two circles in an Apollonius-like packing problem. This problem was inspired by the art deco design in the transom window above the Shadek Fackenthal Library door on the Franklin & Marshall College campus.
Making Change Efficiently
Jack E. Graver
The Product and Quotient Rules Revisited
Roger Eggleton and Vladimir Kustov
Mathematical elegance is illustrated by strikingly parallel versions of the product and quotient rules of basic calculus, with some applications. Corresponding rules for second derivatives are given: the product rule is familiar, but the quotient rule is less so.
A Generalization of the Parabolic Chord Property
The well-known property of quadratic functions, that the tangents at either end of a chord of a parabola meet in a point aligned vertically with the midpoint of the chord is extended to polynomials of degree d. Given two distinct points on a polynomial of degree d, the Taylor polynomials of degree d – 1 at those points meet in d – 1 points whose mean is vertically aligned with the midpoint of the chord joining the two points. This characterizes polynomials of degree d.
PROBLEMS AND SOLUTIONS
Mathematica in Action (Third Edition)
By: Stan Wagon
Reviewed by: Kent E. Morrison