ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The one-panel trapezoidal bound is attained by x squared

Example

For $f (x) = x^{2}$ on $[0, 1]$ with $N = 1$ , the trapezoidal error exactly attains the bound $(b - a) h^{2} ∥ f^{''} ∥_{\infty} / 12 = 1 / 6$ .

Facts & Assumptions

Given: $f (x) = x^{2}$ on $[0, 1]$ , one panel ( $N = 1$ ), and mesh width $h = 1$ .

[F1]

Composite trapezoidal rule: $T_{N} (f) = h (f (0) / 2 + \sum_{j = 1}^{N - 1} f (j h) + f (1) / 2)$ on $[0, 1]$ when $N = 1$ .

[F2]

Second-derivative error bound for the composite trapezoidal rule: $∣ \int_{0}^{1} f - T_{1} (f) ∣ \leq h^{2} ∥ f^{''} ∥_{\infty} / 12$ for $f \in C^{2} [0, 1]$ .

[F3]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $C^{2}$ means the first two derivatives exist and are continuous.

[F4]

A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: a continuous function on a closed bounded interval is Riemann integrable.

[F5]

Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative: the integral of an integrable derivative is the endpoint difference of its antiderivative.

Verification

1.1givenF1F3algebra

The function is $C^{2}$ , with $f^{''} (x) = 2$ and $∥ f^{''} ∥_{\infty} = 2$ . By [F1], $T_{1} (f) = (f (0) + f (1)) / 2 = 1 / 2$ .

1.2givenF4F5

By [F4], $f$ is integrable. The antiderivative $x^{3} / 3$ and [F5] give $\int_{0}^{1} x^{2} d x = 1 / 3$ .

2.1F2step 1.1step 1.2algebra∎

Therefore $∣ \int_{0}^{1} f - T_{1} (f) ∣ = ∣ 1 / 3 - 1 / 2 ∣ = 1 / 6$ , while the bound in [F2] is $1 \cdot 1^{2} \cdot 2 / 12 = 1 / 6$ . Equality holds, so this example attains the stated error bound.

Depends on

Used by

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, §5.6.1–§5.6.4 (standard reference, not scraped)
Endre Süli, Numerical Computation, course notes (standard reference, not scraped)