Slope is how much it rises per unit. Stated that way, the sentence doesn't teach anything — it becomes a decorated definition. Here it becomes two lengths in the same space, and the statement can be checked with a ruler instead of being accepted on faith. A single panel, and everything on the same axes: the function, the line that touches it, and the rate.
Do: turn on the rate and compare the red leg of the triangle with the height of the red curve at the same x — it is the same length, twice. Close the h and see the secant become tangent. Turn on the area and drag the floor: the x-axis is not special.
Drag the h to the left. With h large there are two points and a
secant — the quotient Δf/h is its slope. As h closes, the two
points become one and the secant becomes the tangent: Δf/h approaches
f ′(x), and you see the two numbers converge on the readout line.
It's Station 4 of C2 — "the line through two points, when the two
become one".
Where the rate crosses zero, the curve turns. In the parabola, the vertex is not a special place on the blue curve — it's where the red crosses zero. Rises before, falls after. It's how Fermat found a maximum without a derivative, in C2 §4.
Where the rate does not exist, the curve has a defect. At the cusp, the rate jumps: the tangent line does not know which to choose, and so there is none. In the jump, the function is not even continuous. The three defects are from Station 2 of C1; continuity at a point is Station 4 of C3.
The white circles on the axis mark Bolzano. Where the function changes sign, it passed through zero — without needing to find where. C3, Station 5.
The constant is what the rate doesn't see. On the constant line, the red curve lies on
zero. Adding a constant to the blue doesn't change anything in the red — that's what ker D
means, and it's the reason for the +C in the integral.
The exponential chases itself. In it the red is a shrunken copy of the blue: the rate is proportional to its own value. No other family does this.
In sin x / x the zero is a hole — 0/0 — and the two walls
appear: between −π/2 and π/2, the curve gets stuck between the cosine and the 1. Both go to 1, and
the middle one has nowhere to escape. It's the Squeeze Theorem, C4 §2, and
the limit that closes the series.
About the last two functions on the list.
seno and sin x / x use trigonometry, which this slice doesn't
teach and doesn't need: what it says — the height below is the slope above — holds
for any curve. They're here because they're good examples, and they stay readable
after the angle branch, the triangle, and the circle. The first seven don't contribute
anything.
Turn on the area and drag the floor. The x-axis is not special: it is the graph of the constant zero function, and the area is measured between the curve and it. Changing the floor, the same operation becomes the area between two curves — and the usual floor is so usual that we forget we chose it, like the canonical basis in linear algebra.
Tangent and area live in the same instrument for a purpose: the Fundamental Theorem says that they are the same thing read in two directions. The area by trapping is Station 2 of C2 — Archimedes' gesture.
Local, self-contained instrument: no network, no external library. Sibling of
desenrolamento.html — that one shows where the number comes from; this,
how the number changes.