You recognize a parabola by eye. But recognizing is not reading, and the shape deceives: two similar curves can come from completely different machines. Here the curve appears nameless, and the only path to the answer is what it does — the rate. And the rate is not ready: you raise it, point by point, with your own hand, until it becomes a second curve made of your measurements.
Do: drag where to measure and the red points will trail behind. measure alone sweeps for you, clear measurements restarts. Adjust the step h: in a family it barely changes the answer — at the cusp and the jump, it changes everything. Only then answer.
There is no number of types of curve, and every catalog is the writer's choice. Memorizing a list of formats is the opposite of reading. What exists is an organizer — what the rate does — and it fits in six lines:
rate zero, constant. Rate always the same and not zero, line. Rate that is a line, parabola. Rate that is a parabola, cubic. Rate proportional to the curve itself, exponential. Rate of the same family, shifted, sine and cosine. And rate that jumps is no family at all: it's a defect.
Because they are the ones the rate doesn't take out of place. A polynomial derives into a polynomial, with the degree dropping by one. The exponential derives into itself. Sine and cosine derive into each other, and return to the start in four steps. A family that the rate would take out of itself would not have a chapter: each calculation would open a new object, and there would be nothing to learn all at once.
The h is the step of the measure: Δf divided by 2h, the
slope of a string centered on the point. In families, changing the step barely changes the
answer. At the cusp, it changes everything: the measured rate rises in a ramp instead of jumping, and the ramp
becomes narrower as the step shrinks — the ramp was from the instrument, not the curve. At the jump, the measured value skyrockets and grows without end as the step shrinks.
This is where measuring stops being enough. Measuring the rate is one thing; proving it is that is another, and it requires the operation that the limit provides. This is the topic of C2 — before the word limit — and the family's station is there.
The slice of the rate takes the opposite path: you choose the name on a button and see the curve and its rate. Here the curve comes nameless, and the name is your conclusion. The two together close the pair — rule that becomes curve, and curve that becomes rule.
The eleven family curves of this instrument were verified in code: classifying each one only by what the measured rate does returns the correct label in the eleven, and the answer does not change between h = 0.03 and h = 0.6. The two with flaws change — it's the difference the sheet wants you to see.
Local instrument, self-contained: no network, no external library. The rate f′ was already declared in the slice of the rate, two before this — here it is only measured.