How do we estimate from unevenly spaced data?
You will be able to: Use each actual interval width in left, right and trapezoidal sums.
How do we estimate from unevenly spaced data?
A sensor records a tap at 0, 1 and 3 minutes. The second gap lasts twice as long as the first. Giving both readings equal time would miscount the water.
A useful starting point: How do rectangles approximate accumulation? →
Words and symbols before equations
- Nonuniform partition
- Subintervals with unequal widths.
- Trapezoid
- A region whose vertical endpoint heights are joined by a straight top.
- Trapezoidal estimate
- Width times the average of the two endpoint heights.
- Tabular data
- Function values given at specific inputs.
What this picture assumes
Original model; readouts are rounded. Readings: t=0,1,3 minutes; rates=2,4,3 L/min. Widths 1 and 2. Straight or flat tops are estimation assumptions, not measured behavior between samples. True integral unknown.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Samples (0,2),(1,4),(3,3): widths 1 and 2 min. L=10 L, R=10 L, T=10 L. True accumulation remains unknown from these data alone.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Suppose the recorded rates are 2, 4 and 3 L/min at 0, 1 and 3 min. The widths are 1 and 2 minutes, respectively.
A left sum is 1×2+2×4=10 L; a right sum is 1×4+2×3=10 L. Equal answers do not prove either equals the true integral.
The trapezoidal sum is 1(2+4)/2+2(4+3)/2=10 L. It assumes straight connections between samples for estimation; the actual rate may bend between readings.
Without midpoint measurements or a formula, a midpoint sum cannot be computed from these three endpoints alone. An interpolated midpoint is an extra modeling assumption.
A worked example, step by step
At t=0,2,5 minutes a rate has values 1,3,5 L/min. Find the trapezoidal estimate of addition.
- Use separate widths 2 and 3.
- First trapezoid: 2(1+3)/2=4 L.
- Second trapezoid: 3(3+5)/2=12 L.
- Estimated addition is 16 L. The data alone do not guarantee whether this overestimates or underestimates.
A table does not specify the entire curve between readings. Do not infer an error direction without a stated interval-wide shape condition.
Can the value at t=2 be read directly from this table?
Compare with an explanation
No. Only values at 0,1,3 are supplied; interpolation needs an additional assumption.
Predict. Change one thing. Explain.
Compare left, right and trapezoidal estimates for the fixed 0,1,3-minute readings. Explain why identical numerical estimates do not establish the true area.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Samples (0,2),(1,4),(3,3): widths 1 and 2 min. L=10 L, R=10 L, T=10 L. True accumulation remains unknown from these data alone.
Original model; readouts are rounded. Readings: t=0,1,3 minutes; rates=2,4,3 L/min. Widths 1 and 2. Straight or flat tops are estimation assumptions, not measured behavior between samples. True integral unknown.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the rate units, signed contributions, theorem conditions, domain or antiderivative check. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionRates at t=0,1,4 minutes are 2,6,4 L/min. Compute L, R and T; state one limitation.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Widths are 1 and 3.
- 1 point: L=1×2+3×6=20 L.
- 1 point: R=1×6+3×4=18 L and T=(20+18)/2=19 L.
- 1 point: The intervening curve is unknown, so exact accumulation and error direction are not determined.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Why are unequal gaps important?
Each rate acts across a different time width.
RECALL 2How does T relate to L and R?
It is their average on the same partition.
RECALL 3Does a table prove concavity?
No; additional information is needed between samples.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do we estimate from unevenly spaced data?
- T=Σ (xᵢ−xᵢ₋₁)[f(xᵢ₋₁)+f(xᵢ)]/2.
- Use each gap separately.
- For the same partition, T=(L+R)/2.
Remember: A table does not specify the entire curve between readings. Do not infer an error direction without a stated interval-wide shape condition.
Conditions: Original model; readouts are rounded. Readings: t=0,1,3 minutes; rates=2,4,3 L/min. Widths 1 and 2. Straight or flat tops are estimation assumptions, not measured behavior between samples. True integral unknown.
Refresh Kid · AP Calculus AB Unit 6 · Objectives LIM-5.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.2, LIM-5.A. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. AB scope includes 6.1–6.10 and 6.14. Topics 6.11–6.13 are BC-only; the numbering gap is intentional. Topic 6.14 consolidates the AB antidifferentiation objectives and Skill 1.C. Focused lesson titles, examples and questions are original Refresh Kid material.
Distinguish signed accumulation, unsigned area and initial amount. Error direction needs interval-wide shape information. FTC hypotheses, variable-bound chain factors, integration constants, transformed bounds and domain restrictions are explicit. Bounded jumps are treated separately from differentiability of accumulation. No improper-integral evaluation, integration by parts or partial-fraction decomposition is taught in this AB unit.
The Organic Chemistry Tutor Fundamental Theorem of Calculus Part 1 video title, creator and description were checked; the full video was not reviewed. Khan Academy’s unit destination was checked, but its JavaScript lesson contents were not fully readable by the research tool. OpenStax Sections 5.2, 5.3 and 5.5 were consulted for conceptual cross-checking. No provider scripts, questions, diagrams or artwork were copied. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional spatial inspection and camera controls. The original tank geometry uses existing self-hosted Three.js with its MIT license. Its 2×2 dm base and water depth use the same volume as the rate calculation: 1 dm³=1 L. The initial water is blue and newly accumulated inflow is teal. The rate graph is an abstract amount calculation; it is not the physical shape of water. Camera rotation changes only the view. Full 2D graphs, readouts and explanations remain available without WebGL.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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