SCIENCE · INVESTIGATE AND EXPLAIN

Recover an instantaneous rate from finite secants

Derive the limit and its finite-spacing error before treating a graph slope as instantaneous velocity.

Learning outcome: At t=2 compare h=1,.5 and.1. Derive forward and centered errors and explain which shrinks linearly versus quadratically at leading order.

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

For x=t³−2t at t=2,h=.5, exact v=10, forward secant 13.25 and centered secant 10.25 m/s.

Open the experiment

Predict

Will a centered secant through a cubic equal its derivative at every finite spacing?

Your prediction and reason

Experiment

At t=2 compare h=1,.5 and.1. Derive forward and centered errors and explain which shrinks linearly versus quadratically at leading order.

Compare finite secants with an instantaneous derivative

Worked teaching model · synthetic values · Exact original position x(t)=t³−2t in SI units. Forward secant spans t to t+h; centered secant t−h to t+h. No measurement noise here; h>0, so finite estimates need not equal the analytic derivative.

Evaluation time: 2 s Sampling half-width h: 0.5 s

Reset keeps your prediction, captures and practice history.

Exact position

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.

Exact position: numerical coordinates

time (s)	position (m)
0	0
0.133333	-0.264296
0.266667	-0.51437
0.4	-0.736
0.533333	-0.914963
0.666667	-1.03704
0.8	-1.088
0.933333	-1.05363
1.06667	-0.919704
1.2	-0.672
1.33333	-0.296296
1.46667	0.22163
1.6	0.896
1.73333	1.74104
1.86667	2.77096
2	4
2.13333	5.44237
2.26667	7.1123
2.4	9.024
2.53333	11.1917
2.66667	13.6296
2.8	16.352
2.93333	19.373
3.06667	22.707
3.2	26.368
3.33333	30.3704
3.46667	34.7283
3.6	39.456
3.73333	44.5677
3.86667	50.0776
4	56

Local secant endpoints

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.

Local secant endpoints: numerical coordinates

time (s)	position (m)
1.5	0.375
2.5	10.625

Calculated values under the declared model
Quantity	Value	Unit
Analytic instantaneous velocity	10	m/s
Forward secant	13.25	m/s
Centered secant	10.25	m/s
Forward truncation error	3.25	m/s
Centered truncation error	0.25	m/s
Instantaneous acceleration	12	m/s²

A derivative is a limiting rate; a nonzero sampling interval still carries approximation error.

Expanding the cubic gives forward velocity = 3t²−2+3th+h² and centered velocity = 3t²−2+h². Halving h quarters the centered truncation term, but this exact-data statement does not include measurement error.

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

A derivative is a limiting rate; a nonzero sampling interval still carries approximation error.

Observation to attach to a capture

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

Understand

Define the limiting operation

Average velocity over a nonzero interval h is [x(t+h)−x(t)]/h. Instantaneous velocity is the limit as h approaches zero when that limit exists. A tangent is a local rate, not a line from the origin to a point. Acceleration is the derivative of velocity, not position divided by time squared.

Expand an original cubic

For x(t)=t³−2t, coefficients carry the SI units needed to produce metres. Expanding x(t+h) gives a forward secant 3t²−2+3th+h². The derivative is the limit 3t²−2. The finite error is 3th+h², which depends on both the evaluation time and the sampling interval.

Compare a centered estimate

The centered secant uses [x(t+h)−x(t−h)]/(2h). Odd symmetry cancels the term linear in h, leaving 3t²−2+h². Centering improves the exact-function truncation order, but h² is not zero at a finite h. At t=2 and h=.5, the analytic velocity is 10, forward secant 13.25 and centered secant 10.25 m/s.

Link a second derivative

Differentiating velocity gives a(t)=6t. Thus at t=2 the acceleration is 12 m/s². The positive acceleration says velocity is increasing locally; it does not state that position itself is 12 m or that speed always increases for all t. If velocity is negative, positive acceleration can initially reduce speed.

Test a changed case

For x=2t³+t, v=6t²+1, giving 7 m/s at t=1. Derive the centered error 2h² rather than copying h² from the original cubic. A measured trace introduces reading errors and clock assumptions absent here; later sampling work will show why reducing h can amplify those errors.

Public numerical transfer practice

Try a new practice case

For x(t)=2t³+t, find instantaneous velocity at t=1 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.

Why does a small exact-function truncation error not certify the precision of actual measured samples?

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: Differentiate a vector trajectory and distinguish its directions Next: Reconstruct motion with two integration constants

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.