# The College Mathematics Journal - September 2003

### Articles

A New Wrinkle on an Old Folding Problem
Greg N. Frederickson
The problem about taking a sheet of material, cutting out the corners, and folding up the sides to make a box with maximum volume appeared in Isaac Todhunter 1852 calculus text and in Henry Dudeney's puzzle column in 1903. This paper gives a survey of the problem and then improves on the usual solution by 10%.

Calculus, π , and the Machine Age
Sarah Jane Colley
In 1995 a new formula for π was found that allowed for the rapid computation of its hexadecimal digits (the trillionth is 8). Here is a derivation of it, with some comments on the behavior of computer algebra systems.

A Survey of Online Mathematics Course Basics
G. Donald Allen
There is a movement of some mathematics courses from the classroom to the internet. The author discusses how this has been and can be done.

Generalization of the Arithmetic-Geometric Mean Inequality and a Three Dimensional Puzzle
Hidefumi Kaatsura
It is possible to pack twenty-seven a by b by c boxes in a cube of side a + b + c. In the course of trying to pack a five-dimensional cube (it's known that it can be done in four dimensions) the author found an inequality that allowed him to solve a different three-dimensional problem.

Using Differentials to Bridge the Vector Calculus Gap
Tevian Dray and Corinne A. Manogue
In the world's first calculus textbook (L'Hôpital, 1696) there is not a single derivative, only differentials. The authors advocate bringing differentials back where they belong for evaluating line and surface integrals.

On Generalizing the Pythagorean Theorem
It is known that the squares on the legs and hypotenuse of a right triangle can be replaced with triangles and it will still be thus that the sum of the two smaller areas is equal to the largest one. The authors demonstrate this, and then go further, even to circles.

Tangent Lines of a Conic Section
Daniel Wilkins
The tangent lines to parabolas have some nice properties (see the May 2001Journal, page 194-196). So do tangents to ellipses and hyperbolas, and they characterize those curves. So, we have yet another definition of conic section.

For What Functions is f -1 = 1/f (x) ?
Sharon MacKendrick
A natural question, and an answer is, "Not many." However, there are some, and here is an account of a search for them.

Fallacies, Flaws, and Flimflam
Edward Barbeau, editor

Classroom Capsules
Warren Page, editor

Maximizing the Area of a Quadrilateral
Thomas Peter
The quadrilateral with sides a, b, c, and d that has the largest area is the one whose vertices lie on a circle. Not surprising, perhaps, but how do you prove it?

Predicting Sunrise and Sunset Times
Donald A. Teets
A fairly simple approximation giving good results (unless you live near a pole, North or South) for the times of sunrise and sunset.

A Calculation of ∫0e-x2dx
A new way of getting at the integral of the title, by finding a volume in two different ways.

A Simple Introduction to e
Pratibha Ghatage
A geometric method for defining a fundamental constant of nature.

A Surface Useful for Illustrating the Implicit Function Theorem
Jeffrey Nunemacher
It's z = x 3 + y 3 - 3 xy ( z = 0 is the folium of Descartes), which has two interesting singularities.

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

Media Highlights
Warren Page, editor

Book Review
By Hardy Grant, of Back From Limbo by Carl Huffman