CauchyMittelwertsatz

The Cauchy Mean Value Theorem, also known as the Mean Value Theorem for derivatives, is a fundamental result in calculus. It states that 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 point c in the interval (a, b) such that the derivative of f at c is equal to the average rate of change of f over [a, b]. Mathematically, this is expressed as:

f'(c) = (f(b) - f(a)) / (b - a)

This theorem is a generalization of the Mean Value Theorem for integrals, which states that if a function f is continuous on the closed interval [a, b], then there exists at least one point c in the interval (a, b) such that the integral of f over [a, b] is equal to f(c) times the length of the interval. The Cauchy Mean Value Theorem is a powerful tool in calculus, with applications in various fields such as physics, engineering, and economics. It is also used to prove other important results in calculus, such as the Fundamental Theorem of Calculus.