What do corners and equal endpoint heights change?
You will be able to: Distinguish a failed hypothesis from a failed conclusion and apply Rolle’s special case.
What do corners and equal endpoint heights change?
A V-shaped track-height graph has equal heights at its ends but a sharp corner at its bottom. There is no horizontal tangent along either straight side. Smoothness is the missing condition.
A useful starting point: Why must an instantaneous rate sometimes match an average rate? →
Words and symbols before equations
- Hypothesis
- A condition needed before a theorem gives a guarantee.
- Rolle’s theorem
- The zero-secant-slope case of the MVT when f(a)=f(b).
- Corner
- A point whose finite one-sided slopes disagree.
- Inconclusive
- The theorem cannot settle the claim from these conditions.
What this picture assumes
Original model; numerical readouts are rounded. Both functions are continuous and have equal endpoint values. Only x² is differentiable at zero. A failed hypothesis removes the guarantee, not all possible conclusions.
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.
- Endpoint secant slope=0. At x=0, the derivative is 0. Both hypotheses hold; the derivative matches zero at c=0.
- 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 f(x)=abs(x) on [−1,1], continuity holds, but differentiability fails at 0. The endpoint secant slope is 0; all existing derivatives inside are −1 or +1, so no derivative equals that slope.
For f=x² on [−1,1], all MVT hypotheses hold and the endpoint heights agree. Rolle’s theorem gives an interior c with f′(c)=0; here c=0.
If a hypothesis fails, do not conclude that a matching point is impossible. For example, abs(x) on [−1,2] has no such point, but other nondifferentiable functions can still have a matching rate. Failure removes the guarantee, not all possibilities.
A jump also breaks the guarantee. When writing a justification, name the interval and the exact continuity or differentiability condition, rather than just saying “the graph looks smooth.”
A worked example, step by step
Apply Rolle’s theorem to f(x)=x²−4x on [0,4].
- The polynomial meets continuity and differentiability requirements.
- f(0)=0 and f(4)=0, so the secant slope is zero.
- Solve f′(c)=2c−4=0.
- c=2 is inside (0,4), giving a horizontal tangent.
Equal endpoint heights alone do not guarantee a horizontal tangent. A corner or discontinuity may prevent the theorem from applying.
Must a theorem’s conclusion be false whenever one hypothesis fails?
Compare with an explanation
No. The theorem is then inconclusive; examine the specific function.
Predict. Change one thing. Explain.
Switch between x² and abs(x) on [−1,1]. Compare the zero secant slope, possible tangent slopes and differentiability at the middle.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Endpoint secant slope=0. At x=0, the derivative is 0. Both hypotheses hold; the derivative matches zero at c=0.
Original model; numerical readouts are rounded. Both functions are continuous and have equal endpoint values. Only x² is differentiable at zero. A failed hypothesis removes the guarantee, not all possible conclusions.
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 questionCompare x² and abs(x) on [−2,2]: state the hypotheses, secant slope, and any point with derivative equal to that slope.
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Compare with the answer and four-point rubric
- 1 point: Both are continuous with equal endpoint heights, so both secant slopes are zero.
- 1 point: x² is differentiable inside, and 2c=0 gives c=0.
- 1 point: abs(x) is not differentiable at 0.
- 1 point: Its existing derivatives are ±1, so it has no derivative equal to zero; this does not contradict Rolle.
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 is Rolle related to MVT?
It is the equal-endpoint, zero-average-rate case.
RECALL 2What breaks differentiability in abs(x)?
Unequal left and right slopes at 0.
RECALL 3Does failed hypothesis mean false conclusion?
Not necessarily; it removes the guarantee.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What do corners and equal endpoint heights change?
- Rolle: MVT hypotheses plus f(a)=f(b) imply some interior f′(c)=0.
- A failed hypothesis means no theorem-based guarantee.
Remember: Equal endpoint heights alone do not guarantee a horizontal tangent. A corner or discontinuity may prevent the theorem from applying.
Conditions: Original model; numerical readouts are rounded. Both functions are continuous and have equal endpoint values. Only x² is differentiable at zero. A failed hypothesis removes the guarantee, not all possible conclusions.
Refresh Kid · AP Calculus AB Unit 5 · Objectives FUN-1.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.1, FUN-1.B. 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 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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