Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
Mean Value Theorem
Objectives
- Explain the meaning of Rolle’s theorem.
- Describe the significance of the Mean Value Theorem.
- State three important consequences of the Mean Value Theorem.
Summary
- If f ′(x) = 0 over an interval I, then f is constant over I.
- If two differentiable functions f and g satisfy f ′(x) = g′(x) over I, then f(x) = g(x) + C for some constant C.
- If f ′(x) > 0 over an interval I, then f is increasing over I. If f ′(x) < 0 over I, then f is decreasing over I.
Rolle’s theorem
- If f is continuous over [a, b] and differentiable over (a, b) and f(a) = 0 = f(b), then there exists a point
c ∈ (a, b) such that f ′(c) = 0.
Mean Value Theorem
- If f is continuous over [a, b] and differentiable over (a, b), then there exists a point c ∈ (a, b) such that