Calculus - Mean Value Theorem

The Mean Value Theorem of Calculus states that for any function f that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one point c in the interval (a, b) such that the function's derivative at c is equal to the average rate of change over the interval. In simpler terms, the slope of the tangent line at point c will be the same as the slope of the secant line that connects the points (a, f(a)) and (b, f(b)). This theorem is particularly useful in proving properties about functions and for finding values where functions have certain slopes. It has applications in motion problems where we might want to find an instant where an object is traveling at the average speed over a given time interval.