If the appropriate conditions are satisfied, what does the Mean Value Theorem guarantee?
There is at least one point c in the interval (a, b) at which f'(c) = [f(b) – f(a)] / [b – a]
The Mean Value Theorem (MVT) is a fundamental theorem in calculus that guarantees the existence of a point c in the open interval (a, b) such that the slope of the tangent line to the curve at c is equal to the average rate of change of the curve over the interval [a, b].
More specifically, 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 (a,b) such that:
f'(c) = [f(b) – f(a)] / (b – a)
This means that the instantaneous rate of change of the function at some point c is equal to its average rate of change over the interval [a,b]. Geometrically, this guarantees the existence of a tangent line to the curve at some point c that is parallel to the secant line joining the endpoints of the interval [a,b].
The Mean Value Theorem is a valuable tool in calculus because it allows us to make important deductions about the behavior of a function without knowing detailed information about its entire graph. It has many applications in optimization, graph sketching, and related rate problems.
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