Formula for mean value theorem
The Mean Value Theorem is a fundamental result in calculus that relates the average rate of change of a function over an interval to the instantaneous rate of change at some point within that interval
The Mean Value Theorem is a fundamental result in calculus that relates the average rate of change of a function over an interval to the instantaneous rate of change at some point within that interval. The 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 at least one point c in the interval (a, b) where the instantaneous rate of change (the derivative) equals the average rate of change (the slope of the secant line).
The formula for the Mean Value Theorem is as follows:
If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists a point c in (a, b) such that:
f'(c) = (f(b) – f(a))/(b – a)
where f'(c) denotes the derivative of the function f(x) at the point c, and (f(b) – f(a))/(b – a) represents the average rate of change (slope of the secant line) between the points (a, f(a)) and (b, f(b)) on the graph of the function.
In simpler terms, the Mean Value Theorem guarantees the existence of a point in the interval where the instantaneous rate of change (slope of the tangent line) is equal to the average rate of change over that interval. This theorem is central to many concepts in calculus, such as finding critical points, determining whether a function is increasing or decreasing, and understanding the behavior of functions.
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