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Choose derivative spacing using an error budget

A smaller finite difference can amplify measurement error even while its truncation error falls.

Learning outcome: Compare three spacings at one error bound, then zero versus nonzero error. Retain the error terms separately and choose a grid spacing by the combined bound.

Starting recap

Read derivatives as local rates and integrals as signed accumulation. Review power and chain rules, vector components, units and initial conditions where used.

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Try

At h=.5 and epsilon=.05, the noiseless estimate12.25 has endpoint-error bound .10, giving12.15–12.35 m/s and total absolute error bound .35 m/s.

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Predict

At h=.2 and epsilon=.05, the endpoint-error contribution is bounded by:

Your prediction and reason

Experiment

Compare three spacings at one error bound, then zero versus nonzero error. Retain the error terms separately and choose a grid spacing by the combined bound.

Balance derivative truncation against bounded noise

Worked teaching model · synthetic values · Synthetic x(t)=t³ m with t in seconds, derivative evaluated at2 s using x(2±h). Each measured endpoint can independently differ by at most epsilon m. Bounds are worst-case feasible errors, not Gaussian confidence. Exact cubic scale coefficient1 m/s³ is supplied.

Half-spacing h: 0.2 s Independent endpoint position-error bound: 0.05 m

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Guaranteed error envelope for this cubic

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: h²+epsilon/h (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Guaranteed error envelope for this cubic

h²+epsilon/h

half-spacing h (s)	absolute error upper bound (m/s)
0.1	0.51
0.11125	0.4618148
0.1225	0.4231695
0.13375	0.3917208
0.145	0.3658526
0.15625	0.3444141
0.1675	0.3265637
0.17875	0.3116718
0.19	0.2992579
0.20125	0.2889488
0.2125	0.2804504
0.22375	0.2735277
0.235	0.267991
0.24625	0.2636847
0.2575	0.260481
0.26875	0.2582731
0.28	0.2569714
0.29125	0.2565004
0.3025	0.2567955
0.31375	0.2578016
0.325	0.2594712
0.33625	0.2617629
0.3475	0.2646411
0.35875	0.2680744
0.37	0.2720351
0.38125	0.2764991
0.3925	0.2814448
0.40375	0.2868531
0.415	0.2927069
0.42625	0.2989911
0.4375	0.305692
0.44875	0.3127972
0.46	0.3202957
0.47125	0.3281774
0.4825	0.3364332
0.49375	0.3450549
0.505	0.3540349
0.51625	0.3633664
0.5275	0.373043
0.53875	0.383059
0.55	0.3934091
0.56125	0.4040884
0.5725	0.4150925
0.58375	0.4264172
0.595	0.4380586
0.60625	0.4500133
0.6175	0.4622779
0.62875	0.4748494
0.64	0.487725
0.65125	0.500902
0.6625	0.5143779
0.67375	0.5281506
0.685	0.5422177
0.69625	0.5565773
0.7075	0.5712276
0.71875	0.5861668
0.73	0.6013932
0.74125	0.6169052
0.7525	0.6327014
0.76375	0.6487805
0.775	0.6651411
0.78625	0.6817821
0.7975	0.6987022
0.80875	0.7159004
0.82	0.7333756
0.83125	0.7511269
0.8425	0.7691534
0.85375	0.7874542
0.865	0.8060285
0.87625	0.8248754
0.8875	0.8439943
0.89875	0.8633844
0.91	0.8830451
0.92125	0.9029756
0.9325	0.9231756
0.94375	0.9436442
0.955	0.964381
0.96625	0.9853855
0.9775	1.006657
0.98875	1.028195
1	1.05

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
True derivative	12	m/s
Noise-free central estimate	12.04	m/s
Truncation bias	0.04	m/s
Endpoint-error contribution bound	0.25	m/s
Worst-case total absolute error bound	0.29	m/s
Lowest possible measured estimate	11.79	m/s
Highest possible measured estimate	12.29	m/s

Smaller spacing reduces truncation but amplifies independent endpoint errors.

Expand (2+h)³−(2−h)³ and divide by2h: the noiseless estimate is12+h². Endpoint errors contribute (eplus−eminus)/(2h), bounded by epsilon/h. This is a worst-case sum, not a standard deviation. A common identical additive offset cancels, unlike independently bounded differences. The optimal supported grid depends on epsilon; h→0 is not automatically best.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Smaller spacing reduces truncation but amplifies independent endpoint errors.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Choose derivative spacing using an error budget

A smaller finite difference can amplify measurement error even while its truncation error falls.

For a supplied cubic x=t³, the true derivative at t=2 is12 m/s. Approximate it with the symmetric difference [x(2+h)−x(2−h)]/(2h). Expanding the two cubes cancels constant and even terms and yields12+h² for the stated coefficient1 m/s³. The residual h² is the exact truncation bias for this particular cubic, not a floating-point accident.

Suppose each endpoint has an independently bounded position error of at most epsilon metres. The difference of those errors can be as large as2epsilon, so derivative error from them is bounded by epsilon/h. Combining with truncation gives a conservative absolute bound h²+epsilon/h in the declared units. A known common identical offset cancels instead; independence of allowed errors must not be confused with a statistical independence claim.

At h=.2 s and epsilon=.05 m, truncation is .04 m/s and the noise bound is .25 m/s. The reported finite-difference estimate can lie from11.79 to12.29 m/s. Halving h to .1 reduces truncation to .01 but doubles the noise bound to .50, making the worst-case total larger.

For a continuous ideal choice with positive epsilon, differentiating the envelope gives2h−epsilon/h²=0 and h=(epsilon/2)^(1/3) under this fixed cubic coefficient. The slider offers a finite grid, so compare its actual supported points rather than pretending the analytic optimum is always selectable. Real functions require an independently justified bound on higher derivatives.

Changed-case check

Predict before reading the comparison. At h=.5 and epsilon=.05, the noiseless estimate12.25 has endpoint-error bound .10, giving12.15–12.35 m/s and total absolute error bound .35 m/s.

Retained investigation

Compare three spacings at one error bound, then zero versus nonzero error. Retain the error terms separately and choose a grid spacing by the combined bound. Why does more precise arithmetic not remove uncertain endpoint positions?

Source and evidence boundary

Conceptual reference checked30 October2026: OpenStax University Physics Volume1. All prose, examples and models are original. These are bounded supplied mathematical trajectories or synthetic observations, not observed physical experiments. No launch, impact or apparatus procedure is requested. Record assumptions, prior predictions, errors and independent checks; guided success is not independently demonstrated proficiency.

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Why does more precise arithmetic not remove uncertain endpoint positions?

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