Walk slowly, then sit, then run, then sit again. While you do, sketch a distance-vs-time graph of your motion. Each phase has its own shape.
Graph
y = x < 3 ? 0.5·x : (x < 5 ? 1.5 : (x < 8 ? 1.5 + 2·(x - 5) : 7.5))
Walk slow, sit, run, sit — and read it back
The graph above is one such trip: a gentle climb (walking), a flat stretch (sitting), a steep climb (running), then flat again. Every change of activity is a change of slope — and the slope at any moment is your speed.
Your turn
On a distance-time graph, what does a flat horizontal segment mean?
A distance-time graph that curves (instead of straight segments) means your speed is changing smoothly — that's acceleration, and finding the slope of a curve is exactly what calculus was invented for.
Reading the graph
- Slope up — moving forward
- Flat — sitting still
- Slope down — walking back
- Steeper slope — moving faster
The slope of a distance-time graph IS the speed. That's the seed of calculus: a derivative is just a slope.