How to Calculate the Average Rate of Change for a Function

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, 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 the function changes, on average, over that interval. It represents the slope of the straight line that connects the points (a, f(a)) and (b, f(b)).

To calculate the average rate of change, you can use the formula:

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

Here’s a step-by-step example of how to calculate the average rate of change:

1. Identify the function f(x) and the interval [a, b] over which you want to find the average rate of change.
2. Substitute the values of a and b into the function to find f(a) and f(b).
3. Subtract f(a) from f(b), obtaining f(b) – f(a).
4. Subtract a from b, obtaining b – a.
5. Divide (f(b) – f(a)) by (b – a) to find the average rate of change.

For example, let’s say we have the function f(x) = 2x + 3 and want to find the average rate of change on the interval [1, 4].

Step 1: Identify the function f(x) = 2x + 3 and the interval [1, 4].
Step 2: Substitute a = 1 and b = 4 into the function: f(a) = 2(1) + 3 = 5 and f(b) = 2(4) + 3 = 11.
Step 3: Calculate f(b) – f(a): 11 – 5 = 6.
Step 4: Calculate b – a: 4 – 1 = 3.
Step 5: Divide 6 by 3: (f(b) – f(a)) / (b – a) = 6 / 3 = 2.

Therefore, the average rate of change of f(x) on the interval [1, 4] is 2.

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