Instantaneous Rate of Change

Real Analysis · Axiom Academy

How fast is something moving at a single instant? Watch secant lines pivot into the tangent — and the average rate become the exact one. From "how fast on average" to "how fast right now" The speed reading on a car's dashboard answers a strange question: not how far you went over a trip, but how fast you are moving at this exact instant . You can only ever measure speed over an interval — distance divided by time — yet calculus pins down the speed at a single moment. The trick is to shrink the interval until it vanishes, and watch what the average rate settles onto. A car's position is s(t) = 0.3t^2 + 0.5t + 1 . The dashed line joins two points on that curve — its slope is the average rate of change over the interval. Watch the second point slide in toward t = 2 as the gap h shrinks: the secant pivots, and its slope settles onto a single number. As , the average rate stops changing and locks onto the instantaneous rate: the slope of the tangent at t = 2 . Shrink the interval and chase the limit Drag the gap h down toward zero. The dashed secant always joins t = 2 to t = 2 + h — its slope is the genuine average rate over that interval, computed straight from s(t) . Watch the readout: as h shrinks, the average rate closes in on a single value, and the gap to it collapses. The average rate never quite reaches the limit at any finite h — but it gets as close as you like. That target value is f'(2) = 1.70 . Every point has its own instantaneous rate

This is the written version of the interactive lesson above. See the full Real Analysis course.