AP Calculus AB / Analytical Applications of Differentiation
Lesson
Theorems that justify a slope you cannot see
Rolle and the Mean Value Theorem turn a net change into a guaranteed instantaneous rate.
Learning goals
- Check hypotheses and write a Mean Value Theorem conclusion.
- Locate candidates for extrema using f′.
Explanation
If f is continuous on [a, b] and differentiable on (a, b), the Mean Value Theorem says some c in (a, b) has f′(c) equal to the average rate [f(b) − f(a)] / (b − a). The theorem does not find c for you; it guarantees the slope existed.
Critical points live where f′ is zero or undefined, provided they are in the domain of f. They are candidates, not automatically maxes. A sign change of f′ from + to − is a local max. Endpoints of a closed interval can win the global contest even if f′ never zeros there.
Key terms
- Mean Value Theorem. A guarantee that some instantaneous rate on an open interval matches the average rate on the closed interval, given the stated smoothness.
- Critical point. A point in the domain of f where f′ is zero or undefined.
Common mistakes
- Applying MVT to a function with a corner on the open interval.
- Declaring a critical point a maximum without a test or an endpoint comparison.
Practice
Original Marlow Works items. Check the answer explanation after you try.
What is lim (x→3) of (x² − 9) / (x − 3)?
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
f is continuous on [0, 2] but has a corner at x = 1. Why might MVT fail?
Answer. Differentiability on (0, 2) fails, so the theorem’s hypotheses are not met.
The conclusion is not free. The hypotheses are part of the statement.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Calculus AB course page
Official eight-unit framework and 2027 exam updates.
College Board · All rights reserved · accessed 2026-10-01
Open sourceReference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source