Part Time Scholar

I am getting to the age when my son is studying calculus under the tutilege of  a tutor.  Although he is a few years ahead at math in school, I have to admit the very thought of him doing derivatives makes me feel old.  There is also nostalgia however, as I rekindled my interested in calculus by writing this piece -- and yes, I needed a memory jog from the Internet to complete the formulas.   Most people meet calculus as a set of rules.  Differentiate this.  Integrate that. Memorize formulas. Apply them until the answers come out correctly.  It feels almost mechanical, like a toolkit you learn to use without ever opening it to see how it was built.    But calculus is not really about formulas. At its core, it is about something much more fundamental: how change actually works when you zoom in close enough. The idea of first principles is simple. Instead of accepting shortcuts or memorized rules, you rebuild the concept from the most ba...

I used to pay little attention on boarding at the airport because of my elite status with various airlines. But retirement has kept me grounded and almost swiped my record clean in recent years. Recent trips, including one to Colombia with an infamous American airline, had me thinking more about the science behind plane boarding.   Anyone who has boarded an airplane enough times has likely experienced the strange choreography that unfolds before takeoff. Passengers stand up almost immediately when boarding begins, lines begin forming long before their assigned group is called, and eventually a slow procession starts moving toward the aircraft. For a few moments the line appears to flow smoothly. Then everything stops. Someone is lifting luggage into an overhead compartment. Another passenger realizes they are seated twenty rows further back. A family begins negotiating seating arrangements while the rest of the line quietly waits. Movement resumes briefly before another i...

There is a familiar ritual that unfolds on highways almost every day. You are driving in moderate traffic when a neighboring lane begins moving slightly faster than yours. At first, the difference is subtle. A few cars pull ahead, creating the impression that a better option exists just a few meters away. Almost instinctively, your attention shifts. You begin comparing speeds, scanning gaps, and calculating whether a lane change might save time. Eventually, convinced that an opportunity has appeared, you switch lanes with quiet confidence. For a brief moment, the decision appears justified. The new lane moves smoothly and the cars around you continue forward. Then something strange happens. The lane slows. Vehicles begin compressing together. Meanwhile, the lane you just abandoned suddenly starts moving. Cars you previously passed now drift ahead while you sit wondering whether traffic possesses a strange sense of humor. Experiences like this are common enough that many drivers jokin...

There is a small ritual that almost everyone performs at supermarkets. We scan the checkout area, count the number of people in each line, glance at shopping carts, and make a quick calculation in our heads. The logic seems obvious: fewer people should imply less waiting. Yet many of us have experienced the strange frustration of watching a longer line move faster while ours becomes trapped behind a price check, a failed barcode scan, or a customer searching endlessly for a wallet. At first glance this feels like bad luck, but mathematics suggests something more interesting may be happening. Most people treat supermarket checkout as a simple optimization problem. You arrive, observe a few lines, and choose the one with the smallest number of people. The assumption is intuitive: fewer customers should imply shorter waiting time. However, this ignores the fact that queueing systems are not governed by headcount alone, but by stochastic service dynamics. In queueing theory, the system is...