Finding the Average Rate of Change of a Function: Step-by-Step Example

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

To find the average rate of change of a function f(x) on the interval [a, b], we use the following formula:

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

Here, f(a) represents the value of the function at the beginning of the interval (a), and f(b) represents the value of the function at the end of the interval (b)

To find the average rate of change of a function f(x) on the interval [a, b], we use the following formula:

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

Here, f(a) represents the value of the function at the beginning of the interval (a), and f(b) represents the value of the function at the end of the interval (b). The difference in the function values is divided by the difference in the x-values.

Let’s work through an example to better understand this concept:

Example:

Consider the function f(x) = 3x^2 – 2x + 1 on the interval [2, 5]. We want to find the average rate of change of this function on this interval.

Step 1: Determine the function values at the interval’s endpoints.
f(2) = 3(2)^2 – 2(2) + 1
= 12 – 4 + 1
= 9

f(5) = 3(5)^2 – 2(5) + 1
= 75 – 10 + 1
= 66

Step 2: Calculate the average rate of change.
Average rate of change = (f(5) – f(2))/(5 – 2)
= (66 – 9)/(5 – 2)
= 57/3
= 19

Hence, the average rate of change of the function f(x) = 3x^2 – 2x + 1 on the interval [2, 5] is 19.

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