I saw a wonderful exchange on twitter yesterday on a problem posted by Alexander Bogomolny:
At first this problem didn’t really jump off the page as a good first year calculus example, but then I as the solution that Nassim Taleb posted:
I’m a tiny bit time constrained this morning and can’t get the Taleb tweet to embed right, so here’s the solution a second time just in case the embedding remains broken:
So, Taleb reduces the difficult-looking limit and sums to two integrals. The ideas underlying this reduction are both beautiful on their own and fundamental in calculus.
A few questions that I think would be worth discussing with calculus students are:
(1) [this one was discussed in the twitter thread] Why did the integrals start at 0, and does that matter?
(2) Why is ratio of the integrals equal to the ratio of the sums? This answer to this question is related to the answer to (1). It is also an excellent way to reinforce some of the main ideas behind Riemann sums.
(3) Probably less mathematically interesting, but a good challenge exercise for students is evaluating Taleb’s integral formulation of the problem using l’Hospital’s rule. I say “less mathematically interesting” because you have to evaluate the same integrals in both approaches, but the approach via l’Hospital’s rule allows you to discuss the Fundamental Theorem of Calculus and also review the chain rule. The arithmetic here requires you to be extra careful, but I think the other ideas outweigh the annoying arithmetic.
Too bad that the school year is over – but this is a great example to keep in your back pocket for next year’s calculus classes!