Rolle's Theorem and Mean Value Theorem

Front

What is Rolle's Theorem?

Back

Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.

Front

What are the conditions for applying the Mean Value Theorem?

Back

The Mean Value Theorem can be applied if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b).

Front

State the Mean Value Theorem.

Back

The Mean Value Theorem states that if a function f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Front

What does it mean for a function to be continuous?

Back

A function is continuous if there are no breaks, jumps, or holes in its graph over the interval.

Front

What does it mean for a function to be differentiable?

Back

A function is differentiable at a point if it has a defined derivative at that point, meaning the tangent line exists.

Front

Can a function be continuous but not differentiable? Give an example.

Back

Yes, a function can be continuous but not differentiable. An example is f(x) = |x| at x = 0, where the graph has a sharp corner.

Front

What is the significance of horizontal tangent lines in Rolle's Theorem?

Back

Horizontal tangent lines indicate points where the derivative is zero, which is a key conclusion of Rolle's Theorem.

Front

If a function has three x-intercepts, what can we infer about its derivative?

Back

If a function has three x-intercepts, it must have at least two points where the derivative is zero, according to the Intermediate Value Theorem.

Front

What is the relationship between critical points and the Mean Value Theorem?

Back

Critical points are where the derivative is zero or undefined, and the Mean Value Theorem guarantees at least one critical point exists between any two points on a differentiable function.

Front

Explain the difference between a local maximum and a local minimum.

Back

A local maximum is a point where the function value is higher than all nearby points, while a local minimum is where the function value is lower than all nearby points.