The integral formula for secant is often introduced in a very artificial way. In this post, I look at the history of this integral and give several derivations.
This post discusses two classic problems in analytic number theory: the Gauss circle problem and the Dirichlet divisor problem.
These problems are known to be related at a deep level, a fact which is often missed at first glance because the obvious/early attacks on them look quite different.
In this post, I compare these “trivial” estimates, and show how Gauss’ estimate can be realized using a few different techniques.
For many calculus students, Riemann sums are those annoying things that show up in the derivation of the arc length formula.
In truth, these handy sums have done so much more. In this post, I’ll give some examples of Riemann sums dating from before the birth of calculus and some applications of Riemann sums that are still used today.
Back in high school, I came across the following contest problem – “What is the largest product of a set of positive integers totaling 20?”
It’s a fun problem, so don’t rush past the spoiler tags too fast. In this post, we’ll give the solution to this problem and discuss a “continuous” version of this question. Namely, what happens when we’re allowed to include real numbers in our product?
In differential calculus, the product rule is both simple in form and high in utility. As such, it is typically presented early on in calculus courses, and the proof given is almost always the same.
In this post, we’ll explore the merits of a second proof of the product rule using properties of the logarithm, one that I hope presents a motivated and compelling argument as to why the product rule should look the way it does.