Topic 10.12 Lagrange Error Bound: bound the error of a Taylor polynomial approximation using the Lagrange form of the remainder (BC).

What this topic is asking

The College Board (Topic 10.12, BC only) gives the Lagrange error bound, the general way to bound how far a Taylor polynomial $P_n(x)$ can be from the true function value $f(x)$. Unlike the alternating-series bound, it works for any Taylor approximation, using a bound on the next derivative.

The bound

Reading the bound as the next term

The cleanest way to remember the Lagrange bound is that it is the next Taylor term in absolute value, with the unknown derivative $f^{(n+1)}(a)$ replaced by a guaranteed upper bound $M$. The degree-$n$ polynomial's first missing term is $\frac{f^{(n+1)}(a)}{(n+1)!}(x - a)^{n+1}$; bounding $|f^{(n+1)}|$ by $M$ over the interval turns this into $\frac{M}{(n+1)!}|x - a|^{n+1}$. This is why every ingredient is "$n+1$": the next derivative, the next factorial, the next power. Holding this picture in mind prevents the frequent slip of using $n$ instead of $n+1$ somewhere.

Finding M, the real work

The harder step is choosing $M$, an upper bound on the size of the $(n+1)$-th derivative on the interval between the center and the point. For functions whose derivatives are bounded by a constant, this is easy: every derivative of $\sin x$ or $\cos x$ has absolute value at most $1$, so $M = 1$ always works. For $e^x$, the derivative is $e^x$, which is increasing, so on $[0, c]$ the maximum is $e^c$, and you bound it by a convenient number (as the worked exam question bounds $e^{0.5}$ by $2$). You do not need the exact maximum, only a valid upper bound, and a slightly generous $M$ still gives a correct (if looser) error bound. Justifying the choice of $M$ is what earns the reasoning marks.

A worked error bound

When to use Lagrange versus the alternating bound

The two error tools of Unit 10 cover different situations. The alternating series error bound (Topic 10.10) is simpler but applies only when the series you are truncating is alternating and satisfies the alternating series test; the bound is just the first omitted term. The Lagrange bound is general: it works for any Taylor polynomial, alternating or not, but requires you to bound a derivative. A good strategy is to use the alternating bound when the Taylor series happens to alternate at the point in question (often the case for $\sin$, $\cos$, $\ln(1+x)$), and the Lagrange bound otherwise, or whenever the problem explicitly says "Lagrange error bound." Both give a valid (possibly different) bound; the alternating one is usually tighter and faster when it applies.