November 2003 Contents
Solids in Rn 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
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
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 areh units apart then their circumferences differ by 2ph 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 x4 + 2x3 + 8x2 + 6x + 15 = 0 are ± 1 ± 2i 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.
Edited by Warren Page
Finding the Tangent to a Conic Section Without Calculus
Sidney H. Kung
The title says it all.
A Hairy Parabola
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
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
Edited by Warren Page
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.