Mean Value Theorem
If f is continuous on [a,b] and differentials on (a,b), then there is a number c such thatf'(c)=(f(b)-f(a))/(b-a)
The Mean Value Theorem is a central theorem in calculus that relates the slope of a function to the average rate of change of the function. Simply put, the Mean Value Theorem states that for a differentiable function on an interval, there exists at least one point in the interval where the instantaneous rate of change (slope) equals the average rate of change.
In more technical terms, the Mean Value 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 a point c in (a,b) where the derivative of the function, f'(c), equals the slope of the secant line connecting the endpoints of the interval, (f(b) – f(a))/(b – a).
This theorem is very useful in calculus because it allows us to calculate the instantaneous rate of change of a function at a specific point, without having to resort to finding the limit of the difference quotient. It is also used to prove several important theorems in analysis, including the Fundamental Theorem of Calculus.
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