From d54167c3c56043eb7efd006b3cc409006654ee83 Mon Sep 17 00:00:00 2001 From: Alex Dowling Date: Tue, 18 Aug 2026 14:36:13 -0400 Subject: [PATCH] Fix sign on the Taylor remainder term in the trapezoid derivation 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 --- notebooks/07-dev/Intro-and-Newton-Cotes.ipynb | 2 +- notebooks/07/Intro-and-Newton-Cotes.ipynb | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/07-dev/Intro-and-Newton-Cotes.ipynb b/notebooks/07-dev/Intro-and-Newton-Cotes.ipynb index 04bb50ae..6430d963 100644 --- a/notebooks/07-dev/Intro-and-Newton-Cotes.ipynb +++ b/notebooks/07-dev/Intro-and-Newton-Cotes.ipynb @@ -1366,7 +1366,7 @@ "\n", "Additionally,\n", "\n", - "$$\\int\\limits_a^b \\frac{(x-a)^2}{2} f''(a)\\,dx = -\\frac{h^3}{6} f''(a) = O(h^3).$$\n", + "$$\\int\\limits_a^b \\frac{(x-a)^2}{2} f''(a)\\,dx = \\frac{h^3}{6} f''(a) = O(h^3).$$\n", "\n", "When we plug this into the original integral we get\n", "\n", diff --git a/notebooks/07/Intro-and-Newton-Cotes.ipynb b/notebooks/07/Intro-and-Newton-Cotes.ipynb index ce86dae9..763c969f 100644 --- a/notebooks/07/Intro-and-Newton-Cotes.ipynb +++ b/notebooks/07/Intro-and-Newton-Cotes.ipynb @@ -1263,7 +1263,7 @@ "\n", "Additionally,\n", "\n", - "$$\\int\\limits_a^b \\frac{(x-a)^2}{2} f''(a)\\,dx = -\\frac{h^3}{6} f''(a) = O(h^3).$$\n", + "$$\\int\\limits_a^b \\frac{(x-a)^2}{2} f''(a)\\,dx = \\frac{h^3}{6} f''(a) = O(h^3).$$\n", "\n", "When we plug this into the original integral we get\n", "\n",