Every Term Is Built on the Centre

You'll learnto build a Taylor polynomial one coefficient at a time, to say what the factorial is actually for and to watch it do that job, to keep the derivative order, the factorial and the exponent equal in every term, to read a higher degree as a wider match rather than a better one, and to treat the centre as part of the object rather than as a detail of the wording.

A Taylor polynomial matches a function's value and its first few derivatives at a single point. That point is the centre, and it appears twice in every term: once inside the derivative being evaluated there, and once inside the power it is multiplied by. Change the centre and you have a different polynomial, approximating the same function somewhere else.

8 STEPS · 6 QUICK CHECKS · A HIGHER DEGREE BUYS WIDTH, NOT EXACTNESS · v1

the derivative order, the factorial and the exponent are one number

Before you start

What you're looking at

One plane. The pink curve is cos x throughout, and every violet curve is a polynomial approximating it. The centre is marked in cyan, and unless a step says otherwise it is x = 0.

The scale is deliberately uneven

A unit of y is drawn about 1.5 times taller than a unit of x, so the gaps between a polynomial and the curve are big enough to see. Every claim in this file compares two such gaps at the same x, and that comparison survives the stretch. Nothing here claims a length, an angle or an area.

Watch for

In step 3 the factorials are removed from the coefficients and the readout keeps checking the derivatives at the centre. Watch it report each one as exactly k factorial times too big — the number the factorial was there to cancel.

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