Rates of changeslopes in the wild

0:03Speed hides in steepness

An entire car trip in one curve: time along the bottom, distance up the side. Where the curve climbs steeply, miles pile up fast. Where it runs flat, time passes but distance doesn't budge — the car is parked. Speed is never drawn directly. It hides in the steepness.

0:25Slope from two points

Put numbers on one stretch. One hour in, ten miles. Three hours in, fifty. A straight line through those two points rises forty miles over a run of two hours. Forty divided by two: twenty miles per hour, on average. That average is the slope of the line — rise over run. A straight line cutting through a curve is a secant, and its slope is an average rate.

0:54Every rate is a slope

The same move works far beyond road trips. Money in a savings account — the slope is dollars per month. Water filling a tank — liters per second. Temperature, populations, plant height: whenever you ask how fast something is changing, you're asking about a slope.

1:15Averages depend on the window

There's a catch. Twenty was the average over the full two-hour stretch. Measure only the second half instead, and the same car averages twenty-five. Same trip, different window, different number. An average smears out the details — at any given moment the speedometer could read something else.

1:38Speed at one instant

So ask for one instant. Drag the second point onto the first and the run shrinks toward zero — but the rise shrinks in lockstep, and the tilt of the line holds steady all the way. That surviving tilt is the answer. The speed at a single instant is just the steepness of the curve right there. Rise over run, taken to its limit. That's why the speedometer was never drawn — it was the slope all along.