average rate of change (2.1)
slope of secant line
The average rate of change is a mathematical concept that refers to the average rate at which a quantity changes over a given period of time. In mathematical terms, the average rate of change of a function f(x) over the interval [a, b] is defined as:
(Average rate of change) = (Change in y) / (Change in x) = (f(b) – f(a)) / (b – a)
Here, f(b) and f(a) are the values of the function at the endpoints of the interval [a, b], and b – a is the length of the interval between a and b.
For example, if we have a function f(x) = x^2 and we want to find the average rate of change of the function between the interval [1, 3], then we can use the formula as follows:
Average rate of change = (f(3) – f(1)) / (3 – 1)
= (9 – 1) / 2
= 4
Therefore, the average rate of change of the function f(x) = x^2 over the interval [1, 3] is 4. This means that the function increases at an average rate of 4 units per unit of input in the interval [1, 3].
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