Mean Value Theorem(as rates of change) in words
The Mean Value Theorem, in terms of rates of change, states that if a function is continuous on a 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 the instantaneous rate of change of the function at c is equal to the average rate of change of the function over the interval [a, b]
The Mean Value Theorem, in terms of rates of change, states that if a function is continuous on a 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 the instantaneous rate of change of the function at c is equal to the average rate of change of the function over the interval [a, b].
In simpler terms, this theorem tells us that for any smooth and continuous function, there will be a point within the interval where the slope of the tangent line (instantaneous rate of change) is equal to the slope of the line connecting the endpoints (average rate of change). This implies that at some point, the function must have the same rate of change over the whole interval as it does at that specific point.
Geometrically, it means that if you were to graph the function, there will be at least one point where the tangent line is parallel to the secant line between the endpoints. This theorem is a fundamental concept in calculus and allows us to make connections between the average and instantaneous rates of change of a function.
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