Average Rate of Change of f on [a,b]
f(b)-f(a)/b-a
The average rate of change of a function f on the interval [a, b] is defined as the ratio of the change in the value of the function f between the endpoints a and b to the change in the input variable (independent variable) x between these same endpoints.
Mathematically, the average rate of change of f on [a, b] is given by:
average rate of change of f on [a,b] = (f(b) – f(a)) / (b – a)
This formula is derived by using the slope formula of a line that passes through two points (a, f(a)) and (b, f(b)). The denominator (b-a) represents the change in the independent variable x, while the numerator (f(b)-f(a)) represents the change in the dependent variable y.
The average rate of change of a function can also be interpreted geometrically as the slope of the secant line that connects two points (a, f(a)) and (b, f(b)) on the graph of the function.
It is important to note that the average rate of change of a function can be positive, negative or zero. If the function is increasing on the interval [a, b], then the average rate of change is positive. If the function is decreasing on the interval [a, b], then the average rate of change is negative. If the function is constant on the interval [a, b], then the average rate of change is zero.
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