Calculating the Average Rate of Change of 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 the function changes over that interval

The average rate of change of a function f(x) on the interval [a, b] is a measure of how the function changes over that interval. It can be thought of as the average slope of the function over that interval.

To calculate the average rate of change of f(x) on [a, b], you need to find the difference in the function’s values at the endpoints of the interval and divide it by the difference in x-values.

The formula for average rate of change is:

Average rate of change = (f(b) – f(a)) / (b – a)

Here, f(b) and f(a) represent the values of the function at the endpoints b and a, respectively. The difference between these two function values represents the change in the function over the interval [a, b].

Similarly, (b – a) represents the difference in the x-values of the endpoints b and a.

Let’s take an example to illustrate this concept:

Consider the function f(x) = 2x + 1 on the interval [1, 5].

We need to find the average rate of change of this function on the interval [1, 5].

First, we evaluate f(5) and f(1):

f(5) = 2(5) + 1 = 10 + 1 = 11

f(1) = 2(1) + 1 = 2 + 1 = 3

Next, we calculate the difference in the function values:

f(5) – f(1) = 11 – 3 = 8

Then, we calculate the difference in the x-values:

5 – 1 = 4

Finally, we find the average rate of change:

Average rate of change = (f(5) – f(1)) / (5 – 1) = 8 / 4 = 2

So, the average rate of change of f(x) = 2x + 1 on the interval [1, 5] is 2.

This means that, on average, the function increases by 2 units for every 1 unit increase in x-value within the interval [1, 5].

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