Speedometer

Calculus 1 · Axiom Academy

Your dashboard already does calculus. A speedometer reading is the steepness of your distance-vs-time curve at a single instant — the derivative. You glance down and it says 60. What is that number, really? Speed is distance over time — but at a single instant, no time passes and no distance is covered, so the plain fraction = 0 falls apart. Yet your speedometer answers anyway, ten times a second. The trick it pulls is the whole opening idea of calculus, and you can watch it happen. Press play and take a drive. The car rolls down the road while its distance-so-far is plotted as a curve. Keep your eye on the speedometer: the needle is reading the steepness of that curve at the car's current moment. Where the curve climbs steeply, the needle is high; where it eases off, the needle drops. That instant-by-instant steepness is the instantaneous speed. The needle never reads the curve's height (how far you've gone) — it reads the curve's slope (how fast that distance is changing). That slope is the derivative. Pick any instant, read the speed off the slope Drag the time along the trip. At whatever instant you land on, a short straight line is laid right against the curve — the tangent — and the mini speedometer shows that tangent's slope. That slope is the speedometer reading at that instant. Steep stretch, big number; gentle stretch, small number. (Park it at t = 3 h and you are doing a clean 64 mph.)

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