exponential
exponential 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: "series"
show: "compound"
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.
- a base that is not e does not have slope equal to height ×1
- and at base e the slope is the height ×1
- and the gap left is smaller than the last term added ×1
- and the limit of the staircase is e itself ×1
- every partial sum is below e ×1
- more splits is closer to e ×1
- no number of splits reaches e ×1
- the table shown gets e right to three decimals ×1
- the tangent drawn is the curve's own slope ×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.
One point's worth of information
A Taylor series claims that everything a function does, everywhere, is encoded in its behaviour at a single point. That claim is extraordinary, it is often true, and the cases where it fails are the interesting ones.
AnalysisThe curve that is its own slope
There is exactly one shape of exponential curve whose steepness at every point equals its height at that point. The number that produces it is 2.71828…, and it was not chosen for elegance.
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.