# The College Mathematics Journal - January 2003

### ARTICLES

Apportionment and the 2000 Election
Michael G. Neubauer and Joel Zeitlin

The size of the House of Representatives can make a difference in elections. For example, if the House had had 492 members in 2002, Al Gore would have been elected.,though a size of 536 would have caused George Bush to be the winner. All House sizes greater than 655 would have made the election go to Bush.

Examining Continuity, Partial Derivatives and Differentiability with Cylindrical Coordinates
Thomas C. McMillan

The reason why some functions of two variables are continuous but not differentiable at a point (almost always the origin, at least in textbooks - the functions themselves probably do not care) can be made clear using cylindrical coordinates.

Parrondo’s Paradox - Hope for Losers!
Darrell P. Minor

It seems impossible that two coin-tossing games, both unfavorable to the player, could, when played alternately, have a positive expected gain. Paradoxical indeed, but it can happen. I have not yet internalized the result: I still don’t understand it. But it is true, and enlightenment may eventually come.

A Generalization of a Minimum Area Problem
Russell A. Gordon

Almost all calculus students at one time or another find the equation of the line through a point in the first quadrant that minimizes the area of the resulting triangle. This line solves other problems as well, and they generalize to three dimensions.

Baseball’s All-Stars: Birthplace and Distribution
Paul M. Sommers

Baseball can provide any number of data sets on which to practice your statistics. Here is one more sample.

Sets of Sets: A Cognitive Obstacle

Sets are abstract objects and sets whose elements are sets are more rarified still, so they can cause students to gasp and sometimes to suffocate. Recognizing this is the first step towards doing something about it. What is done may not succeed universally, but incremental improvement is better than none.

On the Square Root of aaT + bbT
Dietrich Trenkler and Götz Trenkler

There is no nice formula for the square root of a matrix. Here, though, is a nice formula for the square root of some matrices.

Tugging a Barge with Hyperbolic Functions
W. B. Gearhart and H. S. Shultz

The hyperbolic functions, which seem to be fading out of the elementary calculus curriculum, have more than one natural application. The Gudermannian, which never was able to get more than a toehold in texts, is also worth remembering.

Fallacies, Flaws, and Flimflam
Ed Barbeau, editor

A power series with an asymmetrical radius of convergence, and other new developments.

Classroom Capsules
Warren Page, editor

On the Indeterminate Form 00
Leonard Lipkin

Applying L’Hôpital’s Rule to textbook exercises on the form 00 usually leads to an answer of 1. How unimaginative! Any answer can be obtained.

On the Work to Fill a Water Tank
Robert R. Rogers

When pumping water, we usually think of lifting slabs of fluid. If this makes you uneasy, here is another way to look it.

A Magic Trick from Fibonacci
James Smoak and Thomas J. Osler

Dividing 100 by 89 produces the first five Fibonacci numbers; dividing 10000 by 9899 produces the first 10, and 1000000/998999 gives 15. Something is clearly going on. The something is here made clear.

Lagrange Multipliers Can Fail to Determine Extrema
Jeffrey Nunemacher

Yes, it is as the title states. Is there nothing we can depend on?

Odd-like (Even-like) Functions on (a, b)
Zhibo Chen, Peter Hammond, and Lisa Hazinski

By moving the point of symmetry of odd and even functions away from the origin, we are able to integrate cos(π sin2θ) from 0 to π/2 with ease.

Problems and Solutions
Benjamin Klein, Irl Bivens, and L. R. King, editors

Media Highlights
Warren Page, editor