Separate quadrature error from an uncertain force record

The exact trapezoid-error formula and deterministic error-bound derivation below are supplied numerical enrichment. The core task remains estimating work from force-position evidence, checking units and explaining limitations; these particular error formulas are not asserted as mandatory AP exam content.

Specify the generator and the measured claim

The teacher-known synthetic force is F(x)=2+3x² N from 0 to 2 m. Its exact integral is 12 J. The workbench supplies samples at n equal intervals and compares a trapezoidal estimate with that known answer. These are generated values, not a measurement of a real instrument or an unknown force law.

Construct the weighted sum

With h=2/n, the composite trapezoid uses half weight for endpoints and full weight for interior samples, multiplied by h. It joins adjacent force samples with straight segments. Because this quadratic is convex, those chords sit above the curve and the estimate overstates work. Direct summation gives an exact bias 4/n² J for this generator.

Propagate a bounded reading error

Suppose every force reading has an assigned absolute bound epsilon. All trapezoid weights are positive and sum to the path length 2 m, so the worst-case coherent work error is 2epsilon J. It does not decrease as one over sqrt(n) without a separate justified independent-noise distribution. A common calibration offset can shift every reading together.

Combine distinct uncertainties honestly

A conservative bound relative to this exact generator is 4/n²+2epsilon. Increasing n reduces the numerical truncation term but not the coherent calibration allowance. Distance readings are stipulated exact here. Uncertain position, an omitted interaction or departure from the quadratic generator requires additional analysis rather than being hidden inside the displayed bound.

Design an independent comparison

A velocity-derived kinetic-energy change can provide a separate comparison with a net-force work integral, but only when mass, velocity and force system boundaries match. Retain raw records and calibrations for both methods. Agreement between two outputs computed from the same known generator is not an independent apparatus validation; altered coefficients can test arithmetic transfer but not replace new measurement evidence.

Worked original case

n=4 gives trapezoid 12.25 J and exact bias.25 J. With epsilon=.2 N, coherent reading bound.4 J and total conservative error bound.65 J. At n=8, bias.0625 J leaves total.4625 J.

Investigation

Compare n=2,4,8 at epsilon=.2. Reconstruct the endpoint weights, separate both error terms, and propose a calibration check that more intervals cannot replace. Retain failed and revised estimates.

Public numerical transfer practice

Try a new practice case

With n=10 and epsilon=.1 N, what is the total conservative work-error bound in joules?

Numerical result in the requested units Check this practice result

This public, unsaved check is not a secure assessment. Explain the derivative, integral or boundary in the retained course evidence workspace.

Sources and evidence boundary

Original teaching, values and calculations, checked 30 October 2026 against the official calculus-based Mechanics framework and OpenStax work and kinetic energy. The numerical packets are supplied or generated, not measured apparatus records. No physical launch, machinery, loading or field intervention is requested. Public feedback and virtual captures do not establish independent mastery or supervised laboratory participation.