Fix sign on the Taylor remainder term in the trapezoid derivation - #169
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Fix sign on the Taylor remainder term in the trapezoid derivation#169adowling2 wants to merge 1 commit into
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The third term of the trapezoid-rule derivation is printed as
\int_a^b (x-a)^2/2 f''(a) dx = -h^3/6 f''(a) = O(h^3)
The minus sign is wrong. Antidifferentiating gives f''(a)(x-a)^3/6 evaluated
from a to b, which is +h^3/6 f''(a) for h = b-a > 0. Verified symbolically.
Nothing downstream changes: the term is immediately absorbed into O(h^3), so
the conclusion -- that one panel of the trapezoid rule is third-order accurate,
and O(h^2) once composed -- is unaffected. It is a display error only, but it is
on a page where students are being walked through the algebra line by line.
Fixed in both the -dev source and the published copy so they stay in sync.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
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The problem
notebooks/07/Intro-and-Newton-Cotes.ipynb, in the Taylor-series derivation of the trapezoid rule, printsThe minus sign is wrong. Antidifferentiating gives$f''(a)\frac{(x-a)^3}{6}$ evaluated from $a$ to $b$ , i.e. $+\frac{h^3}{6}f''(a)$ for $h = b - a > 0$ . Verified symbolically with sympy.
Why it is worth fixing anyway
Nothing downstream changes — the term is absorbed into$O(h^3)$ on the same line, so the conclusion that one panel of the trapezoid rule is third-order accurate (and $O(h^2)$ composite) is unaffected.
But this is a page where students are walked through the algebra line by line, and a reader who checks the integral will find it does not come out. That is exactly the reader the derivation is written for.
The change
One character, in two files — the
-devsource and the published copy, kept in sync:notebooks/07-dev/Intro-and-Newton-Cotes.ipynbnotebooks/07/Intro-and-Newton-Cotes.ipynbNo code cells touched, no outputs changed, no other prose modified.
Context
Found while cross-checking numeric-integration material for the Fall 2026 refresh of the optimization course, which draws on this chapter. The same slip appears in at least two other places and may be worth a look separately:
nsa-fall-2019/Lectures/C12-Numeric-Integration/C12-Numeric-Integration.ipynb(markdown cell 56) — co-authored with Ryan McClarren, so worth a word with him firstSeparately, and not part of this PR: the same chapter's exercise "derive the error order for midpoint and trapezoid rules" is stated without giving an answer, which is correct — the midpoint rule is$O(h^2)$ composite, not $O(h)$ , because node symmetry buys a free order that the interpolant-degree heuristic misses.
🤖 Generated with Claude Code