Mean Value Theorem
The Mean Value Theorem is a fundamental concept in calculus
The Mean Value Theorem is a fundamental concept in calculus. It 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 open interval (a, b) such that the instantaneous rate of change (the derivative) of the function at c is equal to the average rate of change of the function over the interval [a, b].
In other words, if you imagine the graph of the function f(x) on the interval [a, b], the Mean Value Theorem guarantees the existence of a point c within that interval where the tangent line at c is parallel to the secant line connecting the points (a, f(a)) and (b, f(b)). The slope of this tangent line is equal to the average rate of change of the function over the interval [a, b].
Mathematically, the Mean Value Theorem can be stated as:
f'(c) = (f(b) – f(a))/(b – a)
where f'(c) is the derivative of the function evaluated at c, and (f(b) – f(a))/(b – a) represents the average rate of change of the function over the interval [a, b].
The Mean Value Theorem has many important applications in calculus, including the ability to find values such as local extrema, and to prove various results such as Rolle’s theorem and the First Fundamental Theorem of Calculus.
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