mean value theorem
The 'mean value theorem' is a crucial concept in calculus that links the average rate of change of a function to its derivative.
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Definition
C1Calculus
(technical, academic)For a continuous function on a closed interval and differentiable on the open interval, there is a point where the derivative equals the average rate of change over the interval.
Example
- According to the mean value theorem, there must be at least one point where the slope of the tangent matches the slope of the secant line between two points.
- The mean value theorem helps in understanding the behavior of differentiable functions over an interval.
C2Statistics
(technical, academic)For a set of data, there exists a value within the range such that the function's value at that point equals the mean value over the entire range.
Example
- The mean value theorem in statistics can help identify a representative value within a dataset.
- Using the mean value theorem, one can find a point that accurately reflects the average outcome of a statistical function.