Generator

taylor

A generator in the analysis library, called 5 times across 1 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

taylor 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

Partial sums of sin xsin x with its Taylor partial sums of degree 1, 3, 5, 9 about zero. Each extra term buys agreement over a wider interval and none of them is right everywhere.-8-6-4-22468-2-112xsin xdegree 9 is out by 4.7e+0 at x = 5.76

show: "radius"

A smooth function whose series stops working1/(1 + x²) with partial sums of degree 4, 12, 28. Inside |x| < 1 more terms improve the fit; outside it they make it worse, and the function is smooth throughout.-2-1.5-1-0.50.511.52-1-0.50.511.5x|x| = 1at x = 1.4: degree 4 is out by 2.54, degree 28 by 8176

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.

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.

The whole library · What the figures prove