Calculus? Freshman college calculus, never took it. Just read a book and started the class on sophomore calculus. Teaching calculus? Been there; done that,

Calculus? Freshman college calculus, never took it. Just read a book and started the class on sophomore calculus. Teaching calculus? Been there; done that, as a math grad student. Easy to teach -- off the back of my hand. Zero time for preparation. Now with D. Knuth's TeX, it would be easy to pass out very nicely polished class notes, examples, exercises, applications, etc. Here's an application that might interest the students. At one time it literally saved FedEx from going out of business. The BoD wanted some revenue projections for the full, planned fleet and business. So, we knew what the revenue was then and what the revenue should be for the full planned fleet. So, the projections were essentially an interpolation. So, there would be growth, and what would drive that growth? Okay, assume the rate of growth is proportional to (1) the number of current customers talking about FedEx and (2) the number of the rest of the target customers hearing the talking. So, for time t, with t = 0 being the present, let y(t) be the revenue per day at time t. So, we know y(0), the current revenue. Let b be the revenue per day of the full, planned fleet and business. Then at time t, the number of customers talking is proportional to y(t) and the number of the rest of the customers listening is proportional to (b - y(t)). So, for some constant of proportionality k, we have that the growth rate in revenue per day is d/dt y(t) = y'(t) = k y(t) (b - y(t)) where we have y(0) and b. So, this is an initial value problem for a first order, linear, ordinary differential equation, but just freshman calculus is plenty to solve it in closed form. Sure, the solution has some exponentials. I'll post the solution later on request! So, one day Senior VP Planning Mike Basch and I picked a value of k that yielded a reasonable graph, and I drew the graph. That was Friday. The BoD meeting was the morning, Saturday, with Mike traveling. At noon I got a call from Senior VP Roger Frock asking if I knew about the revenue projections and could I come to the BoD meeting? I did and did. When I got there, the graph was on a table, and our two representatives of BoD member General Dynamics were standing in the hall with their bags packed. No one was happy. Roger pointed to a few places on the graph, and I used my HP calculator to reproduce the points. Everyone got happy. Later I learned that the graph had been presented to the BoD early in the meeting; the General Dynamics guys asked how the graph was calculated; all the FedEx people at the meeting tried to learn how; near noon the Dynamics guys gave up on FedEx, got plane tickets back to Texas, went to their rented rooms and packed, and as a last resort returned to the BoD meeting and were standing in the hall when I arrived. With my results, they unpacked and stayed. Had they left, FedEx would have ended. So, just a little calculus saved FedEx, and since I was the only one around who knew still knew calculus I was the only one who understood the solution. The projections were a smooth curve that rose slowly, rose more quickly, had an inflection point, rose less quickly, and became asymptotic from below at b. So, the curve was a lazy S. I suspect that the curve is the famous logistic curve also important in, say, the elements in neural networks and has long been seen as a good, first cut description of the growth of new products, e.g., TV sets when they were new. With the way I derived the curve, it is an axiomatic approach to viral growth or word of mouth market growth. There is lots of nice stuff that can be done with calculus. For more with calculus, I would recommend famous texts by Apostol, Rudin, Royden, and Rudin again, through measure theory and then texts by Loeve, Breiman, Neveu, and Chung for the connections with probability. The third edition of Rudin's Priciples does very well with the modern Stokes theorem, exterior algebra of differential forms, and the inverse and implicit function theorems. The first half of Rudin's Real and Complex Analysis does well with measure theory and introductions to functional analysis. The Apostol text is Mathematical Analysis: A Modern Approach to Advanced Calculus -- it's NOT very "modern" but gives a lot of what need to know to read mathematical physics, especially Maxwell's equations. For stochastic processes, Doob, Karatsas and Shreve. For Princeton and its department of Operations Research and Financial Engineering, the more elementary parts of stochastic optimal control via, say, Nemhauser, Dreyfus and Law, Bertsekas and Shreve, Dynkin and Yushkevich, etc.