Understanding Average Rate of Change in Calculus | Definition, Formula, and Interpretation

average rate of change

The average rate of change is a concept commonly used in calculus and it measures the average rate at which a quantity changes over a given interval

The average rate of change is a concept commonly used in calculus and it measures the average rate at which a quantity changes over a given interval. It is calculated by finding the difference in the values of a function at two points and dividing it by the difference in the corresponding input values.

To express it mathematically, let’s say we have a function f(x) and we want to find the average rate of change between two points, (x1, f(x1)) and (x2, f(x2)). The average rate of change, denoted as ARoC, is given by:

ARoC = (f(x2) – f(x1)) / (x2 – x1)

For example, let’s say we have a function f(x) = 2x + 3 and we want to find the average rate of change between x = 1 and x = 3. Plug in the values into the formula:

ARoC = (f(3) – f(1)) / (3 – 1)
= (2(3) + 3) – (2(1) + 3) / 2
= (6 + 3) – (2 + 3) / 2
= 9 – 5 / 2
= 4 / 2
= 2

Therefore, the average rate of change of f(x) = 2x + 3 between x = 1 and x = 3 is 2.

The average rate of change can also be interpreted as the slope of the line connecting the two points (x1, f(x1)) and (x2, f(x2)) on the graph of the function f(x). This means that the average rate of change represents how the output of a function is changing on average over a given interval of the input values.

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