What can derivative tables prove—and what can they only suggest?
You will be able to: Write evidence-based conclusions without extending sparse samples beyond their scope.
BC foundation: Unit 5 shares these analytical differentiation objectives with AB. Check theorem hypotheses, justify derivative signs over intervals, and compare feasible candidates before claiming an optimum. State the local branch when analyzing an implicit relation.
What can derivative tables prove—and what can they only suggest?
Three temperature readings do not prove the temperature rose at every moment between them. Likewise, a few positive derivative samples can miss negative rates between the sampled inputs.
A useful starting point: How do the graphs of f, f′ and f″ fit together? →
Words and symbols before equations
- Sample
- A value at one listed input.
- Interval-wide evidence
- A formula, sign statement or graph description applying throughout an interval.
- Estimated second derivative
- A finite difference of first-derivative values.
- Justification
- The claim, relevant evidence and calculus reason connecting them.
What this picture assumes
Original model; numerical readouts are rounded. Both displayed curves represent possible f′, not f. They share values 1,2,1 at x=0,1,2. Their signs between samples differ; the formulas, not the three-point table, determine those signs.
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.
- The selected possible derivative at x=0.5 is 1.75. Both curves match every displayed sample. This curve is positive throughout [0,2], but the table alone cannot prove that. The graph represents f′, not f.
- 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 f′ is given only at x=0,1,2 with values 1,2,1. These samples alone do not prove f′>0 for all inputs between them, so they do not prove f increases throughout [0,2].
In fact, p(x)=2−(x−1)² is positive on [0,2], but q(x)=p(x)−8sin²(πx) has the same three sample values and is negative at x=0.5. Both can be derivatives of differentiable functions; the samples hide the difference.
If the problem explicitly states f′>0 on (0,2), then the increasing conclusion is justified. If it states f′ changes + to − at c and f is continuous there, a local maximum is justified.
A centered finite difference (f′(c+h)−f′(c−h))/(2h) estimates f″(c); it is not automatically exact. State approximate units and do not infer a rigorous sign from coarse samples without further assumptions.
A worked example, step by step
A table gives f′(1)=3 and f′(1.2)=3.8. Estimate f″(1.1) and state its limitation.
- Use the slope of the secant through the two first-derivative data points.
- The input gap is 1.2−1=0.2.
- The estimate is (3.8−3)/0.2=4 output units/input units².
- Label it approximate; these two samples do not establish the exact second derivative at 1.1.
A finite table is not a continuous sign chart unless the problem supplies that extra information.
Can identical derivative samples hide different signs between them?
Compare with an explanation
Yes. The two curves in the investigation give a direct counterexample.
Predict. Change one thing. Explain.
Switch between two smooth derivative curves with identical values at x=0,1,2. Inspect x=0.5 and explain why the same table supports different monotonic behavior.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
The selected possible derivative at x=0.5 is 1.75. Both curves match every displayed sample. This curve is positive throughout [0,2], but the table alone cannot prove that. The graph represents f′, not f.
| x | f′ sample |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 1 |
Original model; numerical readouts are rounded. Both displayed curves represent possible f′, not f. They share values 1,2,1 at x=0,1,2. Their signs between samples differ; the formulas, not the three-point table, determine those signs.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the relevant derivative signs, theorem conditions, domain or geometric constraint. 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 questionGiven f′(2)=−1 and f′(2.4)=1, estimate f″(2.2). Does the table prove f has a minimum exactly at 2.2?
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Compare with the answer and four-point rubric
- 1 point: The derivative-value change is 1−(−1)=2.
- 1 point: Divide by 0.4 to estimate f″(2.2)≈5.
- 1 point: The endpoint samples do not give f′(2.2) exactly or establish its side signs there.
- 1 point: Thus they do not prove a minimum exactly at 2.2; further information is needed.
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 1What is missing from a few sample values?
Behavior between the samples.
RECALL 2How should a finite-difference result be labeled?
As an estimate unless exactness is justified separately.
RECALL 3What makes a calculus justification complete?
A claim, sufficient evidence and the relevant theorem or sign rule.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What can derivative tables prove—and what can they only suggest?
- f″(c)≈[f′(c+h)−f′(c−h)]/(2h).
- A sampled derivative sign does not establish its sign between samples.
Remember: A finite table is not a continuous sign chart unless the problem supplies that extra information.
Conditions: Original model; numerical readouts are rounded. Both displayed curves represent possible f′, not f. They share values 1,2,1 at x=0,1,2. Their signs between samples differ; the formulas, not the three-point table, determine those signs.
Refresh Kid · AP Calculus BC Unit 5 · Objectives FUN-4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.9, FUN-4.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 5 has twelve official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Existence theorems require their stated hypotheses. Critical numbers must belong to the function’s domain; interior local extrema and included endpoints are handled explicitly. Extrema and inflection candidates need justification, not just a zero derivative. Sparse derivative samples do not establish interval-wide signs. Optimization includes the constraint, feasible domain and global comparison. Implicit-curve conclusions specify the branch and distinguish finite slopes from vertical tangents.
The Organic Chemistry Tutor Mean Value Theorem and Optimization Problems video titles, creator and relevant descriptions were checked; full videos were not reviewed. The free optimization video mentions additional paid material, which is not required here. Khan Academy’s unit destination was checked; its JavaScript lesson contents were not fully readable by the research tool. OpenStax Sections 4.3, 4.4, 4.5 and 4.7 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. The open-box model is original geometry using existing self-hosted Three.js with its MIT license. A 12 cm square sheet loses four equal corner squares. The remaining net folds into an open box; the geometry uses the same lengths as the labeled 2D net. Fold angle is a construction view, not an independent design variable; displayed volume refers to the fully upright box. Camera rotation changes no mathematical values. Complete 2D diagrams 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 BC 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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