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The Mean Value Theorem is a fundamental theorem in calculus that relates the average rate of change of a function over an interval to its instantaneous rate of change at some point within that interval. Specifically, if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Geometrically, the Mean Value Theorem states that there is a point c between a and b where the tangent line to the curve y = f(x) is parallel to the secant line connecting the points (a, f(a)) and (b, f(b)). The term (f(b) - f(a)) / (b - a) represents the slope of the secant line, which is the average rate of change of f over the interval [a, b]. The term f'(c) represents the slope of the tangent line, which is the instantaneous rate of change of f at point c.

The Mean Value Theorem has numerous applications in calculus and other areas of mathematics. It is used to prove other important theorems, such as Rolle's Theorem and Taylor's Theorem. It also provides a way to bound the difference between function values based on information about its derivative. For instance, if |f'(x)| <= M for all x in (a, b), then |f(b) - f(a)| <= M(b - a). This inequality is crucial for error analysis and approximation techniques. The theorem is a cornerstone for understanding the behavior of functions and their rates of change.