# College Mathematics Journal Contents—January 2018

Our Volume 49 opens with the calculus behind determining the equivalence of generic and brand name drugs, highlighting the Lambert W function. Two articles consider the ever-popular topic of fractals: one (along with the cover image) on modifications of the recipe for Koch snowflakes, the other on counting problems related to Menger sponges, especially large ones built with business cards! The issue introduces Classic Book Reviews with George Salmon's 1848 A Treatise on Conic Sections which is also a source for an exploration of area bisecting centers of triangles. Other article topics include that perennially falling ladder, using maximal intervals to prove theorems of set theory and calculus, and an 8-step program for the artistically challenged triple integrator. —Brian Hopkins

Vol. 49, No. 1, pp. 1-80

## ARTICLES

### The Calculus Behind Generic Drug Equivalence

p. 2.

Stanley R. Huddy & Michael A. Jones

To show bioequivalence of generic and brand name drugs, the Food and Drug Administration (FDA) requires a statistical comparison of three pharmacokinetic values that measure aspects of the drugs’ concentrations. These three values are related to calculus. We show that there is good reason why the FDA considers these values, as any two of the three is enough to recover the concentration of the drug over time for an orally taken drug using a single-compartment model. The results hinge on applications of the Lambert W function.

### Proof Without Words: A Sum Computed by Self-Similarity

p. 10.

Yukio Kobayashi

We wordlessly prove a formula for the sum of powers of a fixed integer using self-similarity of certain tree diagrams.

### Designing Koch-Like Curves

p. 11.

Vincent J. Matsko

When the angles are changed in the usual algorithm to generate the Koch snowflake, a wide variety of images can be produced, some of which possess rotational symmetry. Using elementary number theory, we parameterize infinitely many such images and determine many geometrical properties of families of images, such as which orders of symmetry are possible and exactly how many sets of parameters produce an image with a given complexity and symmetry.

### MegaMenger Graphs

p. 20.

Allan Bickle

In 2014, faculty and students at colleges around the world participated in the MegaMenger project to build a model of the Menger sponge, a type of fractal, out of business cards. This model can itself be modeled using graph theory, with each vertex representing a small cube and an edge between two vertices whenever they share a face. We determine the order, size, and partite set sizes of graphs representing different steps of building the Menger sponge and Sierpinski carpet, along with the surface area of the Menger sponge. Many results involve recurrence relations, which we work through in some detail.

### The Centroid as a Nontrivial Area Bisecting Center of a Triangle

p. 27.

Allan Berele & Stefan Catoiu

We advertise a relatively new and little known subject, bisecting envelopes or deltoids, and illustrate its virtues by giving a short, simple proof to a classical theorem of convex geometry, the 1953 result of H. G. Eggleston that the centroid of a triangle is the unique point in the plane such that three lines through it divide the area of the triangle into six equal-area regions.

### Proof Without Words: Rearranged Alternating Harmonic Series

p. 35.

Yajun An & Tom Edgar

We visually compute the sum of a rearranged alternating harmonic series.

p. 36.

Brittany A. Burke, Zach Jackson & Steven J. Kifowit

The related rates falling ladder problem is well-known calculus exercise with a paradoxical twist. We revisit the problem and derive a closed-form solution. We also describe experimental support for the solution and discuss some related paradoxes that may tempt future experimenters.

### Basic Theorems in the Language of Maximal Intervals

p. 41.

Haryono Tandra

We present proofs of several basic set theory and calculus theorems using the concept of maximal intervals. These may be useful in the classroom, as they are simpler and more direct than standard proofs.

### Triple Integrals for the Sketching-Impaired

p. 46.

Wm. Douglas Withers

This article describes a method for setting up triple integrals without sketching a three-dimensional object. Making such a sketch presents a hurdle for many students. I have employed this method in the classroom for some years with good success, even among students previously stymied by triple integrals. Assuming an initial list of equations or inequalities describing the region, the method can be summarized in a few rules for manipulating them.

## Classroom Capsules

### Is a Taylor Series also a generalized Fourier Series?

p. 54.

Wojciech Kossek

Geometrical insights are very helpful when learning about the Fourier series. Instead of focusing on trigonometric or exponential functions, we consider a more general collection of orthonormal functions. We show in an elementary way that a Taylor series can also be considered a special case of the generalized Fourier series, with an orthonormal base of power functions and a properly defined inner product.

### The Rational Approximation of Small Angles

p. 57.

Harvey Diamond

Can you approximate small angles between vectors using only rational operations on the coordinates? In two dimensions, yes, with the right trigonometry function. What about more than two dimensions?

p. 60.

## Classic Book Review

### A Treatise of Conic Sections by George Salmon, Hodges and Smith

p. 68.

Reviewed by Brigitte Servatius

p. 73.