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Contents for November 1999

**Things I Have Learned at the AP Reading**

by Dan Kennedy

Elementary calculus contains subtleties, which sometimes even those who construct problems for the calculus Advanced Placement test can overlook. You may be able to think of an example of a function whose derivative approaches no limit as x approaches zero but has a derivative at zero, but you probably can't think of a function whose derivative at zero is 0 and whose derivative is positive on either side of zero but which doesn't have an inflection point at zero.

**Minimizing Aroma Loss**

by Robert Barrington Leigh and Richard Travis Ng

Every time you open a coffee can, some of the aroma (which smells much better than the coffee tastes) escapes. Naturally, you would like to make the loss as small as possible. The authors, neither of whom is out of high school yet, show how.

**Recounting Fibonacci and Lucas Identities**

by Arthur T. Benjamin and Jennifer J. Quinn

As we all know, there are almost infinitely many identities involving Fibonacci and Lucas numbers (only for binomial coefficients is the number larger) and they can be established in a large number of ways--induction, the Binet formulas, generating functions, determinants, and so on. The authors give another way of looking at them, combinatorically, and show that many can be reduced almost to proofs without words thereby.

**Do Most Cubic Graphs Have Two Turning Points?**

by Robert Fakler

Well, they tend to when teachers draw pictures of them on the board, but no conclusions can be drawn from that. If they could, we would conclude that most lines have positive slope and most parabolas open up. If your intuition tells you that the answer to the question is "yes," it is operating properly. If fact, almost all cubics (in the proper sense) have relative maxima and minima.

**Folding Stars**

by Charles Waiveris and Yuanqian Chen

When can you fold paper to make a star? Not all of the time, nor none of the time. This paper shows when.

**The Effects of a Stiffening Spring**

by K. E. Clark and S. Hill

The coupled-spring problem is a staple of differential equations courses. What would happen if one of the springs got stiffer and stiffer? Intuition tells us that the system would act more and more like one with a single spring. Once again intuition is vindicated, though, as is often the case, the verification is not trivial.

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Fallacies, Flaws, and Flimflam

edited by Ed Barbeau

A proof that both Cauchy and Schwartz had the inequality sign in the Cauchy-Schwartz inequality going in the wrong direction, and other surprising results.

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Classroom Capsules

**From Square Roots to n-th Roots: Newton's Method in Disguise**

by W. M. Priestley

An appealing way to approximate n-th roots.

**Amortization: An Application of Calculus**

by Richard E. Klima and Robert G. Donnelly

An opportunity to construct a mathematical model of a fascinating subject, namely money.

**Reexamining the Catenary**

by Paul Cella

Texts commonly throw up their hands when confronted with a cable hanging from supports at different levels. They need not.

**Second Order Iterations**

by Joseph J. Roseman and Gideon Zwas

If our power to divide were destroyed, all would not be lost: there exist iterative procedures not involving division that calculate quotients. The second-order procedure is better than both the first- and third-order methods.

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Software Reviews

The *New Mathwright Library*, reviewed by Dan Kalman

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Problems and Solutions

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Media Highlights

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Miscellanea

Including who first found the palindrome "sex at noon taxes" and why students should major in mathematics.

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Book Review

*State Mathematical Standards*, reviewed by Mark Saul