AP Calculus BC glossary

Taylor remainder

Also called: Remainder term, Taylor error term

The Taylor remainder is the exact difference between a function and its Taylor polynomial of a given degree, so the function equals the polynomial plus the remainder. You rarely compute it exactly. You bound its size instead, which turns an approximation into a result with guaranteed accuracy.

f(x) = P_n(x) + R_n(x), \qquad R_n(x) = f(x) - P_n(x)

The Lagrange form gives $R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!}(x-a)^{n+1}$ for some $z$ between the centre $a$ and $x$ . Nothing tells you which $z$ , and that is exactly why the remainder gets replaced by a bound built from the largest value of $\left|f^{(n+1)}\right|$ on the interval.

A Taylor series equals the function it came from at precisely the inputs where $\lim_{n \to \infty} R_n(x) = 0$ . For $e^x$ , $\sin x$ , and $\cos x$ that limit is zero for every real number, which is what licenses those three series everywhere rather than only near the centre.

The mistake

Evaluating the next derivative at the centre and calling that the error. The remainder uses $f^{(n+1)}$ at an unknown point between $a$ and $x$ , so the derivative has to be bounded across the whole interval, not sampled at one convenient end of it.

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