Unit 5
Analytical Applications
15–18% AB | 8–11% BCCED Unit 5
Existence Theorems
Mean Value Theorem, Extreme Value Theorem, and critical points.
Mean Value Theorem (MVT)
Statement
If is continuous on and differentiable on , then there exists some in such that:
The instantaneous rate of change equals the average rate of change somewhere on the interval.
Extreme Value Theorem (EVT)
Statement
If is continuous on , then attains both an absolute minimum and an absolute maximum on that interval.
Guarantees the existence of extrema, but does not tell you where they are.
Critical Numbers
Definition
A number in the domain of where:
• , OR
• does not exist
• , OR
• does not exist
Critical points are candidates for extrema, but not every critical point is an extremum.
⚠ Watch Out
Critical point ≠ extremum
Not every critical number produces a max or min. Example: f(x) = x³ has a critical point at x = 0, but it's neither a max nor a min.Where Absolute Extrema Occur
On a Closed Interval [a, b]
Absolute extrema can only occur at:
• Critical points in , OR
• Endpoints and
• Critical points in , OR
• Endpoints and
⚠ Watch Out
Forgetting endpoints
Students find critical points but forget to check x = a and x = b. Always include endpoints!AP® Calculus AB & BC · Unit 5 Overview · Mr. Brantley