Rates of change, from how quickly a winger gets up to speed to how models learn: derivatives and integrals without the fear.
6 parts Beginner to Intermediate
Start at part 1Every time you say a winger shot away from a defender, you're talking about rate of change. That's the heart of calculus, and a few seconds of sprinting are enough to see the derivative at work.
Beginner Part 1
Still accelerating, at full speed, lost a yard of pace. Each is about what's happening at one moment, which is what the derivative measures, and it gives coaches something to train, measure and improve.
Beginner Part 2
Two players can hit the same top speed and look identical for a second, then one pulls clear. The difference is how quickly their acceleration fades, and that is the second derivative.
Beginner Part 3
Speed tells you how fast; integration tells you how far. It's the area under the speed graph, and it's how tracking data turns thousands of speed readings into distance covered and high-speed running.
Beginner Part 4
One run into space, pulled apart and put back together. Position, speed and acceleration are one chain linked by derivatives and integrals, and it's how tracking data turns dots on a pitch into what coaches want to know.
Beginner Part 5
Machine learning models learn by measuring how their error changes as they adjust a setting, then stepping downhill. That's the derivative at work. 25 SPFL seasons show it finding how much of last season's scoring carries into this one.
Intermediate Part 6