SCIENCE · INVESTIGATE AND EXPLAIN

Design a sampling interval with competing error sources

Separate exact-function truncation, reading bounds and untested calibration assumptions.

Learning outcome: At epsilon=.04 compare h=.1,.2,.3,.5,1. Retain endpoint readings, the analytic teacher value and both error terms; choose an interval by the declared bound, not visual smoothness.

Starting recap

Differentiate or integrate with stated units, initial values, event boundaries and evidence limits.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

At h=.2,epsilon=.04, truncation=.04 and reading bound=.20 m/s; absolute-error bound=.24 m/s.

Open the experiment

Predict

Must the smallest sampling interval produce the most accurate derivative estimate?

Your prediction and reason

Experiment

At epsilon=.04 compare h=.1,.2,.3,.5,1. Retain endpoint readings, the analytic teacher value and both error terms; choose an interval by the declared bound, not visual smoothness.

Balance differentiation truncation and reading bounds

Worked teaching model · synthetic values · Teacher generator x=t³−2t, centered at t=2 s. Each endpoint reading independently lies within ±epsilon. A displayed worst-case pair uses lower minus epsilon and upper plus epsilon. Bound epsilon/h combines with exact cubic truncation h²; not a probability/confidence interval or measured instrument.

Sampling half-width h: 0.2 s Bound per position reading: 0.04 m

Reset keeps your prediction, captures and practice history.

Synthetic reading pair

Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.

Synthetic reading pair: numerical coordinates

time (s)	position (m)
1.8	2.192
2.2	6.288

Error bound across available widths

Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.

Error bound across available widths: numerical coordinates

half-width (s)	velocity error bound (m/s)
0.1	0.41
0.2	0.24
0.3	0.223333
0.4	0.26
0.5	0.33
0.6	0.426667
0.7	0.547143
0.8	0.69
0.9	0.854444
1	1.04

Calculated values under the declared model
Quantity	Value	Unit
Lower-time adverse reading	2.192	m
Upper-time adverse reading	6.288	m
Adverse-pair derivative estimate	10.24	m/s
Exact-data centered estimate	10.04	m/s
Teacher analytic velocity	10	m/s
Positive cubic truncation	0.04	m/s
Reading-error bound	0.2	m/s
Estimate-error lower endpoint	-0.16	m/s
Estimate-error upper endpoint	0.24	m/s
Conservative absolute-error bound	0.24	m/s

Reducing the spacing shrinks truncation but amplifies a fixed position-reading bound.

Subtracting two readings each bounded by epsilon gives a worst-case numerator error 2epsilon; dividing by 2h gives epsilon/h. The cubic centered bias is exactly h². These are deterministic bounds, not independent Gaussian errors or proof that the displayed worst-case pair is typical.

Derive before checking. Retain contrasting captures and explain one assumption that could fail; calculated evidence is not actual laboratory work.

Reducing the spacing shrinks truncation but amplifies a fixed position-reading bound.

Observation to attach to a capture

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

Understand

Distinguish a model from an observation

The teacher generator is x=t³−2t near t=2. Its analytic velocity is 10 m/s. A centered finite difference on exact samples has value 10+h², so its known truncation error is positive h² in the supplied SI convention. A real experiment may not know the underlying cubic; here it is supplied so two error mechanisms can be separated.

Bound the endpoint difference

Let each measured position differ from its true value by at most epsilon. The worst-case difference error is 2epsilon, because the upper reading can be high while the lower reading is low. Dividing by the centered time span 2h gives velocity-reading bound epsilon/h. This deterministic bound does not assume normally distributed or statistically independent errors.

Combine without hiding the sign

The estimate-minus-true error lies between h²−epsilon/h and h²+epsilon/h. A conservative absolute-error bound is h²+epsilon/h. The displayed adverse pair takes the lower-time reading down by epsilon and the upper-time reading up by epsilon, attaining the positive endpoint. It illustrates a permitted worst case, not the expected reading or a confidence interval.

Compare sampling decisions

For epsilon=.04 m, h=.1 gives bound.01+.4=.41 m/s. At h=.2 it is.04+.2=.24; at h=.3 it is.09+.13333≈.22333; at h=.5 it is.25+.08=.33. The smallest available interval is not best under this criterion. For the continuous bound h²+epsilon/h, differentiation gives 2h−epsilon/h²=0, so an interior optimum has h³=epsilon/2 in this particular cubic model; available control choices may not hit it exactly.

Transfer and challenge

At h=.25 and epsilon=.02, the reading contribution is.08 m/s. That is not the total error unless truncation is separately zero or bounded. Use a held-out interval to challenge the cubic assumption, retain uncertainty in timestamps if present, and distinguish a shared clock-scale error from endpoint reading error. Neither a smooth curve nor agreement between two estimates proves independent calibration. All records here are generated rather than observed instrument readings.

Public numerical transfer practice

Try a new practice case

For a centered estimate with h=.25 s and each position reading bounded by±.02 m, find the reading-error bound in m/s.

Numerical result in the requested units

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 explanations and cases, checked 30 October 2026 against the official AP Physics C Mechanics framework, OpenStax calculus kinematics and vector kinematics. All numerical records are supplied or generated, not actual laboratory measurements. No physical launch, drop or moving apparatus is requested. Public feedback is practice, not independent certification.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Could agreement with this generated truth certify a real instrument clock or an unknown trajectory shape?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Resolve independent components and an event boundary Next: Locate a center by integrating a nonuniform mass distribution

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.