What is the MeanValue Theorem?
Differential Calculus
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The MeanValue Theorem has nothing to do with crankiness, but it is one of the most important theoretical tools in the calculus. In written terms, it is defined as the following: If f (x) is defined and continuous on the interval [a, b], and differentiable on (a, b), then there is at least one number on the interval (a, b)—or a < c=""><>—such that:
When f(a) = f(b), this is a special case called Rolle’s Theorem, when we know that f (c) will equal zero. Interpreting this, we know that there is a point on (a, b) that has a horizontal tangent.
It is also true that the MeanValue Theorem can be put in terms of slopes. Thus, the last part of the above equation (on the right of the equal sign) represents the slope of a line passing through (a, f(a)) and (b, f(b)). Thus, this theory states that there is a point c ∈ (a, b), such that the tangent line is parallel to a line passing through the two points.