Generator

Secants closing on the tangent to x²

A generator in the analysis library, called 23 times across 8 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

secant is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

Secants closing on the tangent to x². Secant lines through x = 1 and a second point 1.2, 0.8, 0.5, 0.28, 0.12 away, with the slope of each. They approach 2, the derivative there.

Chords that settle, and chords that do not

A point with two slopes

A rectangle grown on two sides

sin x magnified three times about one point

eˣ and its inverse, reflected in the diagonal

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

and the difference quotient closes on u'v + uv' ×1
and the gap as a fraction of the window falls with it ×1
and the rough curve's do not settle on anything ×1
between 1 and 4 mirrored pairs are marked ×1
between 1 and 4 pairs are marked ×1
between 2 and 4 windows are drawn ×1
between 4 and 14 spacings are measured ×1
both sides grow, so the picture is four pieces and not a signed sum ×1
both sides of the rectangle are positive lengths ×1
by the last halving the corner is under a twentieth of the increment ×1
each magnification is between 2 and 20 ×1
each secant is closer than the last ×1
each step magnifies by between 2 and 20 ×1
every secant from the left has slope −1 ×1
every secant from the right has slope 1 ×1
the bisection really inverts the function ×1
the corner's share of the increment falls at every halving ×1
the curve is one the figure knows ×1
the first window is at most 4 wide ×1
the four pieces fill the grown rectangle ×1
the function increases across the window it is drawn in ×1
the function is increasing where it is marked, so it has an inverse there ×1
the function is one the family knows ×1
the function is one the figure knows ×1
the gap to the tangent falls by the square of the magnification ×1
the increment is the two strips and the corner ×1
the last secant is near the tangent ×1
the marked point is inside the window ×1
the pair is one the figure knows ×1
the rectangle is drawn inside the domain of both functions ×1
the secants are drawn at a list of 2 to 24 positive offsets ×1
the smooth curve's chords close on its tangent ×1
the step is a positive length under 1.6 ×1
the step the rectangle grows by is a positive length under 1.6 ×1
the two sides disagree, so there is no limit ×1
the two slopes at a matched pair multiply to one ×1
the view is one the family draws ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Analysis

A curve with a corner at every point

Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.

Analysis

A rectangle grown on two sides

A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.

Analysis

Area is the undoing of slope

Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.

Dynamics

How fast the staircase arrives

The slope at a crossing decides whether an orbit reaches it. The same number decides how fast — and when the slope is zero the arithmetic changes kind, from a fixed factor per step to a doubling of the correct digits.

Analysis

The flat map that fits closest

A derivative is usually met as a number, which works because a line through a point is described by one. In more than one dimension the object that plays the same role is a linear map, and the number was always a one-by-one instance of it.

Analysis

The slope of a single point

A slope needs two points. A derivative is the slope at one. The construction that bridges the gap is a sequence of secants, and the whole difficulty of calculus is in what "the limit of that sequence" is allowed to mean.

Analysis

The slope of the mirror image

Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.

Analysis

The staircase that is not the diagonal

A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.

The whole library · What the figures prove