“The bread and butter of first semester calculus are optimization problems.
In this talk we will start with a classic example: given a rectangular sheet of
metal/paper/cardboard, cut equal size squares from each corner so that the
sides can be folded up to make an open top box. Given the dimensions of
the original rectangle, how big should these squares be in order t maximize
the volume of the box?
We will solve this problem, and then consider the following extension:
suppose the rectangle has integer dimensions. Is it possible to get a box
such that its dimensions are rational? The solution of this problem leads us
to number theory and the solution of a Diophantine equation. We will also
talk about a generalization of this calculus problem that leads to another
Diophantine equation. Along the way we will talk about various moments
in the history of mathematics and questions related to teaching calculus.”