Mean Value Theorem
This original textbook reading covers part of the AP topic. Follow the AP course outline for complete topic instruction and practice.
- Mean Value Theorem
For any interval of a function [a, b] within which the function is both continuous and differentiable, there exists a point c in the interval [a, b] where the slope of the tangent line at c is equal to the slope of the secant line going through a and b.
Example:
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Glossary and deeper reading · 1 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Mean Value Theorem
The Mean Value Theorem connects an average rate on a closed interval to an instantaneous rate at an interior point. It requires continuity on the closed interval and differentiability on its interior. For example, f(x)=x² on [1,3] has average slope 4 and derivative 4 at x=2. The theorem guarantees at least one such interior point but need not identify a unique one.