Allan A. Struthers, Counterexamples to a Weakened Version of the Two-Variable Second Derivative Test. A hypothesis in the second derivative test for determining the nature of a critical point for a function of two variables is that the second partial derivatives be continuous in a neighborhood of the critical point. Simple rational functions are provided as examples to show that, unlike the single variable case, mere existence of the partial derivatives at the critical point is not sufficient in the two-variable case. Kathy Liebars, Tying Up Loose Ends with Probability. Suppose three strings are folded in half and held so that the six ends appear identical.
Students then choose at random three pairs of ends and tie each pair together.
This classroom activity engages students in computing probabilities in an unusual setting, and by comparing the observed classroom frequencies with the theoretical probabilities it can be used to demonstrate the relationship between relative frequencies and probabilities. William Barnier and Douglas Martin, Unifying a Family of Extrema Problems. The problem of maximizing the product of several variables subject to the constraint that their sum is a given constant, or the dual problem of minimizing the sum subject to the product being fixed, is seen to underlie several calculus book chestnuts. Examples are problems on finding the minimal cost of fencing a rectanglular field of a given area, or the minimal cost box of a given volume, or even the shape of a rectangular building that minimizes heat loss from its walls, floor and ceiling. Symmetry hidden in the solutions of the special problems is exposed when the problems are seen as instances of the more general dual problems.