Four Parts, and the One With No Shortcut

You'll learnto assemble the Lagrange bound from its four parts, to keep the derivative order, the factorial and the power one step past the degree, to find M as a maximum over the whole interval rather than at its ends, and to tell which of the unit's two error bounds a given problem lets you use.

A Taylor polynomial is close to its function near the center and gets worse as you move away. The Lagrange bound says how much worse it can get, and it is built from four parts: a derivative one order past the degree, the size of that derivative across the whole interval, a factorial, and a power. Three of those move together and the fourth is where the marks are lost. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · A MAXIMUM OVER AN INTERVAL · v2

M is the largest the next derivative gets anywhere on the interval

Before you start

What you're looking at

A function plane. The pink curve is the function, the teal one is its Taylor polynomial, and the vertical distance between them at a chosen x is the error you are trying to put a ceiling on.

The question

A polynomial that is exact at the center is wrong everywhere else. How wrong can it get by the time you reach a particular x?

Step 1 / 12