Topic 10.12 key terms
- error
- How far an approximation is from the true value: |true value − approximation|. Usually unknown.
- error bound
- A number the error can never be bigger than, computed without knowing the true value: a ceiling on the error, not the error itself.
- partial sum
- S_n, the sum of the first n terms of a series; the series' sum S is the limit of the partial sums.
- alternating series error bound
- If a series alternates, its term sizes b_n decrease and b_n → 0, then |S − S_n| ≤ b_{n+1}: the error is at most the first term left out.
- overestimate / underestimate
- For an alternating series whose term sizes decrease to 0: if the first term left out is positive, S_n ≤ S (an underestimate); if it is negative, S_n ≥ S (an overestimate). Strict when the sizes strictly decrease.
- Lagrange error bound
- For the degree-n Taylor polynomial of f centered at a: |f(x) − P_n(x)| ≤ M|x − a|^(n+1)/(n+1)!, where M ≥ |f^(n+1)(t)| for every t between a and x.
- M (the derivative ceiling)
- Any number at least as big as |f^(n+1)(t)| for every t between the center a and x. It must cover the whole interval, not just the center; it need not be the smallest such number.
Transcript
325 words · 1 parts
T+60
It’s the same night and the same rocket from last lesson, a little less than a minute into its climb. At exactly sixty seconds we measured four things, its height, climb rate, acceleration and jerk, and from those four numbers alone we forecast where it would be ten seconds later. We know that forecast was good because we let the rocket fly first and checked, and it came through less than a meter away. In real life, nobody gets to check the future before it happens.
If we had to choose a window at seventy seconds that this rocket could never miss, how narrow would you dare make it? What you’d really want is a promise, like this green band, which says that wherever the rocket really is at seventy seconds, it can’t be anywhere outside it. Look how tight it is, less than a meter above or below the forecast, a narrower window than the rocket is wide.
There it goes, sixty-five centimeters from the forecast, right inside the promise. By the end of this lesson, you’ll be able to make promises like that yourself, for a rocket, a lighthouse lamp and your calculator’s sine button, before you ever see the right answer. The BC exam asked for exactly that in 2023, 2024 and 2025.
Before we build a promise like that, let’s get clear on what we mean by how wrong. Every approximation sits some distance from the true value, and that distance is the error, the absolute value of the true value minus the approximation. We almost never know the error. If we did, we’d just fix the approximation, so what we compute instead is a bound, a number the error can never be bigger than.
Keep those two apart in your head all lesson. The error is the real gap, and the bound is the ceiling over it. Mixing them up is exactly what costs students the point on the exam.