AP Calculus BC glossary

Lagrange error bound

Also called: Taylor remainder

The Lagrange error bound limits how far a Taylor polynomial can be from the function it approximates. It uses the maximum size of the next derivative on the interval between the centre and the point of interest.

∣Rn(x)∣≤max⁡∣f(n+1)(z)∣(n+1)!∣x−a∣n+1\left|R_n(x)\right| \le \frac{\max\left|f^{(n+1)}(z)\right|}{(n+1)!}\left|x-a\right|^{n+1}

It has the shape of the next term in the series, with the derivative replaced by its largest possible value on the interval. That substitution is what turns an unknown remainder into a computable bound.

For sine and cosine every derivative is bounded by one, so the maximum is simply one and the bound reduces to a factorial expression.

It is a bound, not the error

The actual error is usually much smaller. Questions ask you to show the error is less than some amount, which the bound does even when it is loose.

Appears in: Unit 10: Infinite Sequences and Series (BC)