The Mean Value Theorem is a fundamental theorem in calculus that states that for a continuous and differentiable function on a closed interval, there exists at least one point within that interval where the slope of the tangent line is equal to the average rate of change of the function over the interval.
The Mean Value Theorem is used in calculus to prove various theorems and to solve problems related to finding extreme values, evaluating integrals, and understanding the behavior of functions. It provides a powerful tool for analyzing the behavior of functions on a closed interval.
The Mean Value Theorem holds when the function is continuous on a closed interval and differentiable on the open interval. Additionally, the slope of the tangent line at the endpoints of the interval must be equal to the average rate of change of the function over the interval.
The Mean Value Theorem has a geometric interpretation that states that for any continuous and differentiable function on a closed interval, there exists at least one point within that interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints of the interval.
Rolle's Theorem is a special case of the Mean Value Theorem where the average rate of change of the function over the closed interval is equal to zero. This means that there exists at least one point within the interval where the slope of the tangent line is equal to zero, or in other words, where the function has a stationary point.