What does the Mean Value Theorem guarantee?
The Mean Value Theorem (MVT) is a fundamental theorem in calculus that guarantees the existence of at least one point in the interval (a, b) where a function is continuous and differentiable, such that the instantaneous rate of change (slope) at that point is equal to the average rate of change over the entire interval
The Mean Value Theorem (MVT) is a fundamental theorem in calculus that guarantees the existence of at least one point in the interval (a, b) where a function is continuous and differentiable, such that the instantaneous rate of change (slope) at that point is equal to the average rate of change over the entire interval.
In more precise terms, let’s say we have a function f(x) that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). The MVT states that there exists a number c in (a, b) such that:
f'(c) = (f(b) – f(a))/(b – a)
In simple terms, this means that at some point within the interval (a, b), the instantaneous rate of change of the function (represented by the derivative f'(x)) is equal to the average rate of change of the function over the interval (represented by the slope of the secant line between the points (a, f(a)) and (b, f(b)).
The MVT is significant because it provides a direct link between the derivative of a function and its average rate of change, which has numerous applications in calculus. It allows us to conclude certain properties about a function based on the behavior of its derivative, and vice versa.
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