Speedometer Mathematics

Calculus 1 · Axiom Academy

When your speedometer reads a number at one instant, what exactly is it measuring? The answer is a slope that comes alive when you shrink time to nothing. The paradox hiding inside a speedometer Speed is distance over time: how far divided by how long . But a speedometer reads a number at a single instant — and in an instant, no time passes at all. Divide a distance by zero time and the formula falls apart. So how can a needle possibly point at "instantaneous" speed? Track a car whose position follows s(t) = t² + 2t . Pick a moment t and look a little later, at t + Δt. The line through those two points on the position graph — the secant — has slope equal to the average speed over that stretch. Watch what happens as Δt shrinks toward zero: the second point slides back and the secant pivots, settling onto the single line that just grazes the curve. That limiting line is the tangent , and its slope is the speed at that one instant. As Δt → 0 the secant stops being an approximation and becomes the tangent — average speed turns into instantaneous speed, exactly what the needle reads. Shrink Δt and watch the number settle That pivoting secant has an exact value at every step — the difference quotient , average speed over [t, t + Δt]. Fix the moment at t = 2 and drag Δt down toward zero. The chord shortens, its slope is computed live, and the running value homes in on one number it never quite reaches by arithmetic but reaches in the limit.

This is the written version of the interactive lesson above. See the full Calculus 1 course.