Mean Value Theorem The Mean Value Theorem states that for any function defined and continuous on the closed interval [a, b], there exists a number c in...
Mean Value Theorem
The Mean Value Theorem states that for any function defined and continuous on the closed interval [a, b], there exists a number c in (a, b) such that:
f'(c) = (f(b) - f(a))/(b - a)
This means that the average rate of change of the function on the interval is equal to the instantaneous rate of change of the function at some point in the interval.
Examples:
At any point c in the interval (0, 1), the average rate of change is equal to the instantaneous rate of change, which is 2. Therefore, c = 0 is the point where the mean value theorem holds.
At any point c in the interval (1, infinity), the average rate of change is equal to the instantaneous rate of change, which is -1. Therefore, c = infinity is the point where the mean value theorem holds.
At any point c in the interval (-1, 2), the average rate of change is equal to the instantaneous rate of change, which is 6. Therefore, c = 2 is the point where the mean value theorem holds