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November 2003 Contents

**Solids in R**^{n} Whose Area is the Derivative of the Volume

Michael Dorff and Leon Hall

The derivative of the volume of a sphere with respect to its radius is its surface area, and the derivative of the area of a circle with respect to its radius is its circumference. Here we see when else that happens.

**A Serendipitous Proof**

David Perkins

Two methods give two different formulas for the center of the circle inscribed in a right triangle. When set equal to each other, they do not give something trite, as "1 = 1", but the Pythagorean theorem.

**Constructing a Poincaré Line with Straightedge and Compass**

David Hecker

The Poincaré disk models a non-Euclidean geometry, but in it geodesics are arcs of circles. Here is how to find them using only Euclidean tools.

**The Rationals are Countable - Euclid's Proof**

Jerzy Czyz and William Self

It's not really Euclid's of course, but it enumerates each rational exactly once: 0, 1, 1/2, 2, 3/2, 1/3, 2/3, 3, 5/2, 4/3, 5/3, 1/4, 2/5, 3/4, 3/5, 4, … . You could never guess how that order arose.

**Dice Distributions Using Combinatorics, Recursion, and Generating Functions Janet M. **

McShane and Michael I. Ratliff

Three different ways to find the chance of getting a sum of 17 when five dice are thrown. Take your choice!

**The Band Around a (non) Convex Set**

Jack Stewart and Annalisa Crannell

That result that if two concentric circles are*h* units apart then their circumferences differ by 2p*h* also holds if the curves aren't circles, but any convex curve. Here we see that the curves don't even have to be convex.

**A Rational Root Theorem for Imaginary Roots**

Sharon Barrs, James Braselton and Lorraine Braselton

Using this nice extension of the rational root theorem, you can see almost at a glance that the only possible rational complex roots of *x*^{4 }+ 2*x*^{3} + 8*x*^{2} + 6*x* + 15 = 0 are ± 1 ± 2*i* and ± 2 ± *i* .

**When Equalities are not Equal: Missing Mathematical Precision in Teaching Texts, and Technology**

Michael J. Bossé amd N. R. Nandakumar

Algebra texts, teachers, and calculators all say some things are equal when they are not. They really shouldn't.

**Fallacies, Flaws, and Flimflam**

Edited by Ed Barbeau

When you pluck a string, you naturally think of trigonometric functions. Sometimes you should think of parabolas.

**Classroom Capsules**

Edited by Warren Page

**Finding the Tangent to a Conic Section Without Calculus**

Sidney H. Kung

The title says it all.

**A Hairy Parabola**

Aaron Montgomery

That calculus problem about a group of people going on a boat ride (or something) with the cost per person decreasing as the number of riders increases illustrates the dangers of using continuous approximations to discrete problems.

**An Improved Remainder Estimate for Use with the Integral Test**

Roger B. Nelsen

Sharper bounds for *f*(*n* +1) + *f*(*n* + 2) + *f*(*n* + 3) + … in terms of the integral of *f*(*x*).

**A Modified Discrete SIR Model**

Jennifer Switkes

A model for the spread of an epidemic assumes that each sufferer is infected for the same number of days. This can be improved on.

**Maximal Revenue with Minimal Calculus**

Byron L. Walden

What price will maximize revenue? The answer can be found by folding paper.

**Problems and Solutions**

Edited by Ben Klein, Irl Bivens, and L. R. King

**Media Highlights**

Edited by Warren Page

**Miscellanea**

**Book Review**

by Frank Swetz, of *Math Through the Ages: A Gentle History for Teachers and Others*, by William P. Berlinghoff and Berlinghoff and Fernando Q. Gouvêa.