College Mathematics Journal Contents—January 2015

Volume 46 of The College Mathematics Journal joins the MAA100 celebration by featuring new articles from winners of George Pólya Awards. This issue starts with Michael A. Jones (a 2012 recipient) and Jennifer Wilson using calculus to explore an exponent in the formula used in Michigan to determine child support payments. Calculus topics continue with Acosta and Lawson on contractions and Michael Maltenfort revisiting a rotated graphs of functions. History of mathematics is the other major theme of this issue; Thomas Bannon follows Hamilton's letters on the process of discovering the quaternions while Cadeddu and Lai explore Fermat's methods to find extrema, and Dominic Kylve reviews Joseph Mazur's book on the history of mathematical notation. Four Proofs Without Words are joined by Zsolt Langvársky's three new visual proofs of the Pythagorean theorem. —Brian Hopkins

Vol 46 No 1, pp 1-80

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ARTICLES

Adjusting Child Support Payments in Michigan

Michael A. Jones and Jennifer Wilson

For divorced parents in Michigan, the base monetary support each parent is expected to contribute to raising their child is adjusted according to the number of (over)nights spent with the parents. Curiously, this adjustment is based on a rational polynomial function with an exponential parameter that has changed over the years. We use calculus to explain the effect of changing this parameter, prove an equivalent formulation by applying the derivative/antiderivative relationship, and apply this formulation for cases in which the total level of base support is fixed but there is a change in the amount contributed by each parent. We also compare the method used in Michigan to a fixed-cost/marginal-cost model.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.3

Proof Without Words: The Area of a Regular Dodecagon

Roger B. Nelsen

We determine the area of a regular dodecagon inscribed in a unit circle by dissecting it into twelve triangles that are reassembled into three squares.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.10

What Distributes Over Exponentiation?

Sherman Stein

Multiplication distributes over addition and exponentiation distributes over multiplication. What distributes over exponentiation? We answer that question, assuming continuity and then not. We also demonstrate the collaborative development of proofs.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.11

Maxima and Minima Without Derivatives?

Lucio Cadeddu and Giampaolo Lai

Almost fifty years before Leibniz and Newton developed the tools of calculus, Pierre de Fermat was able to solve standard calculus problems. We examine several applications of his methods (providing additional details), offer some additional exercises, and briefly consider Fermat's place in the development of the derivative.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.15

Proof Without Words: The Maximum Sum of Inradii

David Richeson

Using the Japanese theorem on cyclic polygons, we visually show that the sum of inradii in a triangulation of a regular polygon approaches the diameter of the circumcircle as the number of sides tends to infinity.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.23

Secants, Tangets, Rotations, and Reflections

Michael Maltenfort

Given the graph of a continuous function on an interval, if we know the slopes of all the secant lines, then we can determine which rotations and reflections of the graph are also graphs of functions and, for those that are, whether the new functions are one-to-one. Provided that the original function is differentiable on the interior of the interval, we can determine the slopes of all secant lines by calculating the range of the function's derivative, provided that we know any subintervals where the function is linear.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.24

Weak Contractions and Fixed Points

Daniel Acosta and Terry Lawson

The notation of a contraction is an important concept in analysis. We explore here the weaker notion of a weak contraction in a variety of contexts, involving the reader throughout with questions to provide additional details or proofs, and suggest possible further areas of exploration.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.35

Proof Without Words: A Trigonometric Proof of the Arithmetic Mean-Geometric Mean Inequality

Roger B. Nelsen

We prove wordlessly the arithmetic mean-geometric mean inequality for two positive numbers by an equivalent trigonometric inequality.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.42

The Origin of Quaternions

Thomas Bannon

We discuss a letter that Hamilton wrote a letter the day after he discovered quaternions. Describing what led to his discovery, it gives a valuable insight into the deliberations of a great mathematician as he struggles with a difficult problem.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.43

Proof Without Words: The Pythagorean Theorem

Nam Gu Heo

We give an especially symmetric visual proof of the Pythagorean theorem.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.51

Proving the Pythagorean Theorem by Letting the Sides Vary

Zsolt Lengvárszky

By fixing either one leg or the hypotenuse of a right triangle and letting the other sides vary, we obtain three new proofs of the Pythagorean theorem.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.52

Classroom Capsules

Unexpected Conjectures about -5 Modulo Primes

David Lowry-Duda

When a student was asked to investigate a problem with no expectations or given approach, she made unlikely sounding conjectures that mutated into an unexpected way of recognizing when −5 is a square modulo primes. This paper follows her conjectures and examines their successes and shortcomings.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.56

Grandma Makes Granola

Richard Bedient and Courtney Gibbons

In introductory calculus, most optimization problems begin with a continuous function. Departing from this trend, we describe a plausible micro-economic scenario with a discontinuous cost function. The case of multiple workers requires a more refined analysis than simply taking a derivative.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.1.58