Average Rate of Change of f(x) on [a,b]
f'(x) = Positive
The average rate of change of a function f(x) over the interval [a,b] is given by:
average rate of change = [f(b) – f(a)] / (b – a)
This formula simply calculates the slope of the line connecting two points on the curve of the function, namely (a, f(a)) and (b, f(b)).
Graphically, the average rate of change represents the slope of the secant line that passes through the two endpoints of the interval [a,b]. The secant line joins two points on the curve and provides an estimate of the average rate of change of the function over the interval.
Intuitively, the average rate of change of a function f(x) on an interval [a,b] indicates how quickly the output value of the function changes on average as x moves from a to b. If the average rate of change is positive, the function is increasing; if it is negative, the function is decreasing; if it is zero, the function is constant.
Example: Consider the function f(x) = x^2 – 4 defined on the interval [-2,2]. The average rate of change of f(x) on this interval is:
average rate of change = [f(2) – f(-2)] / (2 – (-2))
= [(2^2 – 4) – ((-2)^2 – 4)] / 4
= [0 – 0] / 4
= 0
Thus, the function f(x) has an average rate of change of zero on the interval [-2,2], which means that it is constant.
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