Which theorem relates the average rate of change to the derivative of a function?

The Mean Value Theorem establishes a crucial relationship between the average rate of change of a function over an interval and its instantaneous rate of change, represented by the derivative at specific points within that interval. According to this theorem, if a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point within that interval where the derivative of the function equals the average rate of change across the entire interval. This means that at that particular point, the slope of the tangent line (derivative) matches the slope of the secant line that connects the endpoints of the interval, effectively demonstrating how the behavior of the function changes on average correlates with its behavior at specific points.

The other theorems listed focus on different aspects of calculus. The Extreme Value Theorem, for instance, deals with finding maximum and minimum values of a function on a closed interval, while Rolle's Theorem is a special case of the Mean Value Theorem that requires the function to take the same value at both endpoints. The Fundamental Theorem of Calculus connects differentiation with integration, establishing the relationship between these two core concepts in calculus, but it does not directly address the relationship between average rate of change and derivatives.