## 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 quickly 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 quickly the function changes over that interval. It can be calculated by finding the difference in the function values at the endpoints of the interval and dividing it by the difference in the values of x at those endpoints.

Mathematically, the average rate of change of f(x) on [a, b] can be written as:

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

Here, f(a) represents the value of the function at the initial point a, and f(b) represents the value of the function at the final point b.

To understand this concept better, let’s consider an example. Suppose we have a function f(x) = 2x + 3, and we want to find the average rate of change of this function on the interval [1, 5].

First, we find the value of the function at the endpoints of the interval:

f(1) = 2(1) + 3 = 5

f(5) = 2(5) + 3 = 13

Next, we calculate the difference in the function values and divide it by the difference in the values of x:

Average Rate of Change = (f(5) – f(1)) / (5 – 1) = (13 – 5) / (5 – 1) = 8 / 4 = 2

Therefore, the average rate of change of f(x) on [1, 5] is 2. This means that, on average, the function f(x) increases by 2 units for each unit increase in x within the interval [1, 5].

It is important to note that the average rate of change provides an overall measure of the function’s rate of change over the interval but does not tell us how the function behaves at any specific point within that interval.

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