The Maximal Deflection on an Ellipse
At each point of an ellipse one can attach a normal vector and a radial vector, the latter defined as the vector from the center of the ellipse. At the ends of the major and minor axes, the two vectors coincide, but at all other points they are separated by an angle . What is the maximum value that can attain, and where does it occur? This question leads to a surprisingly rich area of investigation, encompassing several different analytic solutions, geometric interpretations, an extension to n dimensions, and a geographical application.
Group Testing: Four Student Solutions to a Classic Optimization Problem
This article describes several creative solutions developed by calculus and modeling students to the classic optimization problem of testing in groups to find a small number of individuals who test positive in a large population.
A Non-Smooth Band Around a Non-Convex Region
J. Aarão, A. Cox, C. Jones, M. Martelli and A. Westfahl
The area of a constant-width band is shown to be twice the length of its middle section times its width h. The length of the middle section is shown to be equal to the length of the inner boundary plus πh. These results are obtained using the change of variables formula for double integrals.
More Combinatorial Proofs via Flagpole Arrangements
Duane DeTemple and H. David Reynolds II
Combinatorial identities are proved by counting the number of arrangements of a flagpole and guy wires on a row of blocks that satisfy a set of conditions. An identity is proved by first deriving and then equating two expressions that each count the number of permissible arrangements. Identities for binomial coefficients and recursion relations are obtained.
Fibonacci Identities via the Determinant Sum Property
We use the sum property for determinants of matrices to give a three-stage proof of an identity involving Fibonacci numbers. Cassini's and d'Ocagne's Fibonacci identities are obtained at the ends of stages one and two, respectively. Catalan's Fibonacci identity is also a special case.
Fallacies, Flaws, and Flimflam
Ed Barbeau, editor
Michael Kinyon, editor
Distortion of Average Class Size: The Lake Wobegon Effect
This note investigates average class size from both the teacher's and the student's point of view.
Exhaustive Sampling and Related Binomial Identities
Jim Ridenhour and David Grimmett
If 100 contestants participate in a bicycle race that consists of 10 stages and 20 riders are randomly chosen and tested for performance enhancing drugs after each stage, what is the probability that all the racers are tested at least once? The answer to this question gives rise to a family of identities relating members of Pascal's triangle.
Controlling the Discrepancy in Marginal Analysis
The main feature of this note is a way of choosing nice marginal analysis examples in business calculus.
Stirling's formula Via Riemann Sums
Riemann sums are exploited "to the limit" in this elementary derivation of Stirling's asymptotic formula for n! It is presented in modular form suitable for a classroom presentation or a guided sequence of exercises for students.