Four Parts, and the One with No Shortcut

You'll learnto assemble the Lagrange bound from four parts that all move together, to take the derivative order one past the degree you stopped at, to bound that derivative over the whole interval rather than at either end, to recognise that an endpoint reading can give a number that is not a bound at all, to tell this bound apart from the alternating one, and to know which series the alternating bound is simply not available for.

A Taylor polynomial's error has a formula, and it looks like the next term with its derivative replaced by a bound. Everything in it is one step past the degree you stopped at. The part with no shortcut is that bound, which has to hold across the whole stretch between the centre and the point — and reading it off the ends instead can hand you a number that is not a bound at all.

8 STEPS · 6 QUICK CHECKS · A MAXIMUM OVER AN INTERVAL · v1

the bound has to hold everywhere in between

Before you start

What you're looking at

A function plane, not the index panel the rest of this unit uses. Pink is the function, teal is the Taylor polynomial approximating it, and the vertical distance between them at a given x is the error the approximation makes there.

Step 4 changes what the axes mean

In that one step the curve is not a function being approximated — it is the size of the third derivative, plotted across the interval the bound has to cover. The caption and the legend both say so, and the step after it goes back to the original plane.

Watch for

A bound drawn in red is one that fails: it is shorter than the error it claims to cover. That happens twice, and both times the arithmetic is correct and the setup is not. A unit of y is drawn about half as tall as a unit of x in the cosine window, so do not read curvature off the picture.

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