AP Calculus BC glossary

Taylor polynomial

A Taylor polynomial is the finite piece of a Taylor series obtained by stopping after a chosen degree. It approximates the function near the centre, and higher degree generally means a better approximation over a wider range.

The degree counts the highest power actually present, which can be lower than the number of terms you generated. For sin⁡x\sin x the odd powers alone appear, so the third and fourth degree polynomials are the same expression, both equal to x−x36x - \frac{x^3}{6}.

The error is bounded by the Lagrange error bound, or by the first omitted term when the series alternates and its terms decrease in magnitude.

Watch which derivative

The coefficient of (x−a)n(x-a)^n uses the nth derivative at the centre divided by n!n!. Dropping the factorial is the most common slip in building these.

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