AP Calculus Slides December 5, 2007

The Mean Value Theorem states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists some value c in the interval (a,b) such that the slope of the tangent line at c is equal to the average rate of change of f over the entire interval [a,b]. In other words, the derivative of f at c is equal to the slope of the secant line through points (a, f(a)) and (b, f(b)). The Mean Value Theorem provides a relationship between the derivative of a function at a point and the change in function value over an interval containing that point.