Mean Value Theorem and Rolle's Theorem from "summary" of Differential Calculus by S Balachandra Rao
Mean Value Theorem and Rolle's Theorem are two important theorems in calculus that help us understand the behavior of a function on a given interval. Rolle's Theorem states that if a function f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one number c in the open interval (a, b) such that f'(c) = 0. In simpler terms, if a function takes the same values at two points on an interval, then the function has a critical point in between where the derivative is zero. The Mean Value Theorem is an extension of Rolle's Theorem and states that if a function f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the open interval (a, b) such that f'(c) = (f(b) - f(a))/(b - a). This means that the average rate of change of the function over the interval is equal to the instantaneous rate of change at some point within the interval. Both theorems are based on the concept of the derivative of a function and provide insights into the behavior of functions on given intervals. Rolle's Theorem deals with functions that have equal values at the endpoints of an interval, while the Mean Value Theorem relates the average rate of change of a function to its instantaneous rate of change at a specific point. These theorems are fundamental in calculus and are used extensively in analyzing functions and their properties. By understanding and applying Rolle's Theorem and the Mean Value Theorem, we can gain valuable insights into the behavior of functions and make predictions about their values and derivatives at certain points.
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