riemann
riemann 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.
At its defaults
show: "sequence"
show: "pathological"
show: "accumulate"
What it checks while it draws
Collected by running the family and listening to lib/verify.js, not written
here. The count is how many separate times this build put that claim to the test.
- each panel is at least as close as the one before it ×1
- more bars means less error ×1
- no width, no area ×1
- the accumulation only ever climbs, because f is positive ×1
- the function is one the figure knows ×1
- the panels really are in increasing order of rectangles ×1
- the printed total is the area of the bars drawn ×1
- the quoted exact area is the area under this curve ×1
- the slope of the accumulated area is the height of the curve ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Adding up rectangles until they stop being rectangles
The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.
AnalysisArea 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.
AnalysisThe 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.