How do differentials estimate a small measurement change?
You will be able to: Distinguish the tangent-predicted change from the actual change.
How do differentials estimate a small measurement change?
A square has side 5 cm. Increasing the side by 0.02 cm adds almost 0.2 cm² of area. The small corner square explains the tiny difference between that estimate and the exact change.
A useful starting point: When is a tangent estimate too high or too low? →
Words and symbols before equations
- dx
- The chosen small input change in a differential estimate.
- dy=f′(a)dx
- The tangent-predicted output change at anchor a.
- Δy
- The actual difference f(a+dx)−f(a).
- Relative change
- Output change divided by the original nonzero output.
What this picture assumes
Original mathematical model; readouts are rounded. Square side s=5 cm, A=s². dA=10ds; actual ΔA=10ds+ds². Differential is an estimated change. The displayed remainder is computed exactly from this polynomial.
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.
- ds=0.02 cm. Estimated change dA=0.2 cm²; actual change ΔA=0.2004 cm². Omitted remainder ds²=0.0004 cm²; estimated new area=25.2 cm².
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For A=s² at s=5 cm, let ds=0.02 cm. The differential dA=2s ds=10(0.02)=0.2 cm² estimates the area change.
The actual change is ΔA=(5.02)²−25=0.2004 cm². The extra (ds)²=0.0004 cm² is omitted by the linear estimate.
The estimated new area is A+dA=25.2 cm², not merely 0.2 cm². Differential notation is another way to organize the same tangent-line approximation.
Here dA/A=2ds/s=0.008, or about 0.8%. This is a first-order estimated relative change, not a guaranteed measurement-error bound. A chosen tolerance requires further information.
| Feature | Differential dy | Actual change Δy |
|---|---|---|
| Calculation | f′(a)dx | f(a+dx)−f(a) |
| Meaning | Tangent-predicted change | Actual function change |
| New output | f(a)+dy, approximate | f(a)+Δy, exact |
A worked example, step by step
Estimate the change in y=x³ from x=2 to x=2.01, then compare with the exact change.
- Use anchor a=2 and dx=0.01.
- f′(2)=3·2²=12, so dy=12(0.01)=0.12.
- Actual change Δy=(2.01)³−8=0.120601.
- The linear change is smaller by 0.000601; estimated new value is 8.12.
dy is an estimated change, not automatically the actual Δy or the new value f(a)+Δy.
Is the differential estimate exact for a linear function?
Compare with an explanation
Yes. A linear function has a constant slope and no curvature remainder.
Predict. Change one thing. Explain.
Change the signed side increment ds. Compare the two thin rectangular strips with the exact algebraic remainder (ds)²; explain why the approximation improves as abs(ds) shrinks.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
ds=0.02 cm. Estimated change dA=0.2 cm²; actual change ΔA=0.2004 cm². Omitted remainder ds²=0.0004 cm²; estimated new area=25.2 cm².
Original mathematical model; readouts are rounded. Square side s=5 cm, A=s². dA=10ds; actual ΔA=10ds+ds². Differential is an estimated change. The displayed remainder is computed exactly from this polynomial.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the relevant rate relationship, tangent estimate, or limit argument and its conditions. 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 questionA square side decreases from 4 to 3.98 cm. Find dA, ΔA and the estimated new area.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: ds=−0.02 cm and A=16 cm².
- 1 point: dA=2·4·(−0.02)=−0.16 cm².
- 1 point: ΔA=3.98²−16=−0.1596 cm².
- 1 point: Estimated new area=15.84 cm²; actual area=15.8404 cm².
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 1How do dy and Δy differ?
dy uses the anchor’s tangent slope; Δy uses actual function values.
RECALL 2What makes a differential estimate useful?
A sufficiently small input change within a differentiable local model.
RECALL 3Does a differential automatically bound measurement error?
No. It is a local estimate, not a rigorous error guarantee.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do differentials estimate a small measurement change?
- dy=f′(a)dx; Δy=f(a+dx)−f(a).
- Estimated new value=f(a)+dy.
Remember: dy is an estimated change, not automatically the actual Δy or the new value f(a)+Δy.
Conditions: Original mathematical model; readouts are rounded. Square side s=5 cm, A=s². dA=10ds; actual ΔA=10ds+ds². Differential is an estimated change. The displayed remainder is computed exactly from this polynomial.
Refresh Kid · AP Calculus AB Unit 4 · Objectives CHA-3.F · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 4.6, CHA-3.F. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 4 has seven official topics. Topic 4.7 assesses 0/0 and ∞/∞ quotient forms; other indeterminate forms are excluded from the core lessons. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative units and signs are interpreted in context. Speed is the magnitude of velocity; turning requires a sign change. Related-rate equations hold at nearby times and are differentiated before snapshot values are inserted. Cone and ladder models have explicit physical domains. Tangent approximations remain estimates; error direction requires behavior on the relevant interval. L’Hôpital’s rule requires an eligible quotient form, nearby differentiability, nonzero denominator derivative and an existing finite or infinite derivative-ratio limit. A failed derivative-ratio limit is inconclusive about the original quotient.
The Organic Chemistry Tutor video creators and relevant descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.4, 4.1, 4.2 and 4.8 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model shows the circular water surface and axial cross-section of a tip-down conical tank. The water radius and height obey the same similar-triangle ratio in 2D and 3D. Camera rotation only changes the view; signed flow and height controls describe an instantaneous state. Complete labeled 2D geometry, rates and equations remain available without WebGL.
Independent teacher review and observation of students remain pending. Checks do not certify scientific 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 to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
Want to work through this with a tutor?
Bring your question about How do differentials estimate a small measurement change? Your explanation and answers remain free to access.