Math / How steep
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Change · rate
A hill under your feet. A pile that grows while you watch. The question is not “how high?” alone — it is “how fast does height change as you move along?” Rise per run on the graph is the slope. That rate is the tool school later names (derivata). Purpose first: how steep / how fast it grows. The name comes last.
— a still rectangle. Now the floor grows while you watch.
Two people, same job: how fast a quantity changes. Newton from motion and fluxions; Leibniz from differences and the long s for sum. Priority fights were bitter. The tool is the rate.
Builders already knew steeper roofs shed snow. Merchants watched how fast a stock rose. The graph slope is that everyday steepness, drawn.
Zoom in on a smooth curve: locally it looks like a straight run. The slope of that touch-line is the rate at that spot — same word “tangent” as the shadow table, new job.
Honesty: we show one hill and one rate. We are not dumping a table of differentiation rules.
The curve is height against distance along. Move the marker. The short touch-line’s rise-over-run is the steepness at that spot. Positive = climbing. Zero = flat crest or trough. Negative = going down (lutning = stigning / sträcka).
At x = 2.0 · height ≈ 3.2 · slope ≈ 1.1
Sample hill: height = 0.15 x² · (6 − x) style bump — drag to feel climb, crest, descent. Units are pretend metres; the idea is the rate.
Derivatan f′(x) är lutningen i punkten — hur fort höjden ändras när x rör sig.
Next lesson flips the question: given the rate curve, how much piles up in total? That is area under the rate — the integral.
1. Climb. At one spot the touch-line rises 2 m for every 5 m along. What is the slope? Are you climbing or descending?
Estimate: 2/5 = 0.4 — gentle climb.
Slope = rise/run = 2/5 = 0.4. Positive → climbing. Steepness 0.4 means 0.4 metres up per metre along.
2. Crest. At the top of a smooth hill the touch-line is flat. What is the rate of height change there?
Estimate: flat means rise zero — rate zero.
Slope = 0. Height is not changing with a tiny step along. That is why “derivative zero” shows up at crests and troughs — the rate paused.
Hands-on (drag where you stand) is 20. Each right gate item is 20.
1. What is this tool doing?
2. Slope = 2/5 at a spot means…
3. At a smooth crest the slope is zero. Why does that fit?
4. Which is a real question?