De middelwaardestelling

The Mean Value Theorem establishes a relation between a function's average rate of change over an interval and its instantaneous rate of change at some point within that interval.

The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point c in the open interval where the instantaneous rate of change is equal to the average rate of change between the endpoints a and b.

To relate this to a real-life example, consider a vehicle driving along a road between two points: its velocity is low initially, increases midway, and decreases toward the end.

A tangent line touches the curve at a single point, and the slope of this line gives the instantaneous rate of change of velocity at that moment. The slope of a secant line, which passes through two points on the curve, shows the average rate of change of velocity between those points.

This secant slope equals the overall change in the function's value along the closed interval divided by the interval's horizontal length.

The theorem guarantees that, at least one moment during the trip, the instantaneous acceleration must have been exactly equal to the average acceleration.