Theorem of Mean Value

The Mean Value Theorem states that for a continuous and smooth curve between two points, there is at least one point where the slope of the tangent (the instantaneous rate of change) is the same as the average slope between those two points. In simpler terms, if you travel between two locations on a straight road, there must be a moment when your speed matches the average speed for the entire trip. This concept is fundamental in calculus and helps ensure that a function behaves predictably over intervals.