AP Calculus BC

Error Bound vs Actual Error

The actual error is the real difference between an approximation and the true value, which you can only compute if you already know the answer. An error bound is a guarantee that the error is no bigger than a stated number, computable without knowing the answer. Bounds are what exams ask for.

Actual error

Use when: The exact value is known and the question asks how far off an estimate was.

Error bound

Use when: The exact value is unknown, and you need to show the estimate is accurate to within some tolerance.

Side by side

	Actual error	Error bound
Formula	$\left\|f(x) - P_{n}(x)\right\|$	Alternating: $\left\|a_{n+1}\right\|$ ; Taylor: Lagrange $R_{n}$
Needs the true value	Yes	No
Is it a guarantee	It is the exact gap	An upper limit, usually not tight
Typical question	How far off was the estimate	Show the estimate is within $0.001$
Relationship	Always at most the bound	Always at least the actual error

The last row is the point of a bound: it is allowed to be pessimistic. The alternating series bound often overstates the error by a factor of two or more, and that is fine, because a guarantee that the error is under $0.001$ is exactly what the question asked for.

Which bound to reach for

An alternating series with decreasing terms gets the simple bound: the size of the first omitted term. Anything else uses the Lagrange remainder, which needs a bound on the next derivative over the interval.

Frequently asked questions

What is the difference between actual error and an error bound?

Actual error is the real gap and needs the true value. A bound is a computable guarantee that the gap is no larger than a stated number.

Is the bound usually close to the actual error?

Not necessarily. Bounds are often pessimistic, which is acceptable because they only have to guarantee an upper limit.

Which bound do I use?

For an alternating series with decreasing terms, the first omitted term. Otherwise the Lagrange remainder.

In the CED: Unit 10: Infinite Sequences and Series (BC)