Calculating the Average Rate of Change for a Function: Definition, Formula, and Example

average rate of change of f(x) on [a, b]

The average rate of change of a function f(x) on the interval [a, b] is a measure of how much the function changes, on average, over that interval

The average rate of change of a function f(x) on the interval [a, b] is a measure of how much the function changes, on average, over that interval.

To calculate the average rate of change, we need to find the difference between the values of the function at the endpoints of the interval and divide it by the difference in the x-values of those endpoints.

The formula to calculate the average rate of change is:

Average Rate of Change = (f(b) – f(a)) / (b – a)

Where f(b) represents the value of the function at the endpoint b, f(a) represents the value of the function at the endpoint a, and (b – a) represents the difference in the x-values of the endpoints.

Let’s illustrate this with an example:

Suppose we have a function f(x) defined as f(x) = 3x^2 – 2x + 1, and we want to find the average rate of change of this function on the interval [2, 4].

First, we need to find the value of the function at the endpoints:

f(2) = 3(2)^2 – 2(2) + 1 = 12 – 4 + 1 = 9
f(4) = 3(4)^2 – 2(4) + 1 = 48 – 8 + 1 = 41

Next, we find the difference in the function values:
f(4) – f(2) = 41 – 9 = 32

Finally, we calculate the difference in the x-values of the endpoints:
4 – 2 = 2

Now, we can find the average rate of change:

Average Rate of Change = (f(4) – f(2)) / (4 – 2) = 32 / 2 = 16

Therefore, the average rate of change of f(x) on the interval [2, 4] is 16. This means that, on average, the function increases by 16 units per unit change in x over this interval.

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