Rolle's Theorem Explained and Mean Value Theorem For Derivatives  Examples  Calculus  Summary and Q&A
TL;DR
Rolle's Theorem and the Mean Value Theorem are explained and applied to different functions in this video.
Questions & Answers
Q: What are the conditions for Rolle's Theorem to apply?
The function must be continuous on a closed interval, differentiable on the open interval, and f(a) must equal f(b) for the function on that interval.
Q: How do you find the c value that satisfies the conclusion of Rolle's Theorem?
First, find the derivative of the function. Set the derivative equal to zero and solve for x. Check if the resulting c value is within the open interval. If it is, it satisfies the conclusion of Rolle's Theorem.
Q: How is the Mean Value Theorem different from Rolle's Theorem?
The Mean Value Theorem requires the function to be continuous on the closed interval and differentiable on the open interval, just like Rolle's Theorem. However, it does not require f(a) = f(b). Instead, it states that there exists a c value where the slope of the tangent line is equal to the slope of the secant line between the endpoints.
Q: How do you apply the Mean Value Theorem?
Compute the derivative of the function and evaluate f(a) and f(b) at the endpoints of the interval. Then, use the formula f'(c) = (f(b)  f(a))/(b  a) to solve for the possible c values. Verify if these c values are within the open interval.
Summary & Key Takeaways

Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, with f(a) = f(b), there exists a number c in the open interval where f'(c) = 0.

The Mean Value Theorem states that if a function is continuous on the closed interval and differentiable on the open interval, there exists a number c in the open interval where the slope of the tangent line is equal to the slope of the secant line between the endpoints.