The College Mathematics Journal
November 2001 Contents
Estimating Large Integrals: The Bigger They Are, The Harder They Fall
You probably have never felt the need to evaluate a really big integral, something along the lines of exp(exp(x)) times some large power of x, from 0 to, say, 1,000,000. Even so, we should not let monsters like that roam around untamed. This paper gives an easily-remembered method of subduing them.
CORDIC: Elementary Function Computation Using Recursive Sequences
Using a single family of recursion relations, it is possible to calculate products, quotients, sines, cosines, arctangents, square roots, hyperbolic sines and cosines, logarithms, exponentials, and hyperbolic arctangents. Thats the way computers do it.
Conformality, the Exponential Function, and World Map Projections
Timothy G. Feeman
When Mercator set about representing the sphere of our planet on a sheet of paper in 1596, he decided that it was important that angles be preserved, and so they are in the Mercator projection. Stereographic projection is conformal as well, as is exp(z). We can have conformal projections of hyperboloids too, if we want them.
In Search of a Missing Link: A Case Study in Error-Correcting Codes
A natural progression from including a single parity-check bit in a message to the Hamming-Golay error-correcting code, as found by the authors pre-college students. As always, hindsight shows how natural and inevitable it is.
The Profit in Being Unbalanced
Wolf von Rönik
Currency exchange rates can present opportunities for sure profits, if you can find them and act quickly enough. All you have to do is identify unbalanced matrices. Warning: if you lose your shirt trading dollars, yen, and euros, the Journal will not make good your losses. On the other hand, contributions from accumulated profits are always welcome.
Dipsticks for Cylindrical Storage TanksExact and Approximate
Pam Littleton and David A. Sánchez
Given a horizontal cylindrical storage tank, it would be handy to have a dipstick with marks indicating 1, 2, ... gallons (or 100, 200, ... for bigger tanks). Where to put the marks? Heres how to do it.
Geometric ProgressionsA Geometric Approach
Michael Strizhevsky and Dimitry Kreslavskiy
How to make pretty pictures of some geometric progressions (the first author teaches at the Art Institute of Atlanta) and, incidentally, to verify that cot(a/2) = (sin a)/(1 - cos a).
Fallacies, Flaws, and Flimflam
Edited by Ed Barbeau
Some functions with two different antiderivatives, a proof that the cardinality of the class of subsets [0, 1] is the same as the cardinality of [0, 1], and other items.
The Logarithm Function and Riemann Sums
Riemann sums with unequal subintervals have their uses, among which is showing what happens to n(a^(1/n) - 1) as n increases.
An Application of LHôpitals Rule
A students clever but erroneous application of LHôpitals Rule turns out to be all right after all.
The Attraction of Surfaces of Revolution
What is the force exerted by a massive surface of revolution on a point mass located on the axis of symmetry? Assuming an inverse square law of attraction, of course. Here is the answer.
An Elementary Approach to ex
John W. Hagood
Let e be the unique real number larger than 1 such that xe-x is greatest when x = 1. This unusual definition has advantages.
Why It Might Seem That Christmas Is Coming Early This Year
Time, alas, moves faster as we age. This universal perception can be quantified, though nothing can be done to slow the passage of the years.
Russell A. Gordon
How to make series add up to anything.
Generating Functions and the Electoral College
The chance of having a tie in the electoral vote for president is small enough so that we dont have to worry about it.
Problems and Solutions
Edited by Benjamin G. Klein, Irl C. Bivens, L. R. King, and Todd G. Will
Student Research Project
Jack E. Graver and Lawrence J. Lardy
Enclosing fields with fences is a constant activity in elementary calculus. The problem has several variations that can be investigated.