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

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 values change on that interval

The average rate of change of a function f(x) on the interval [a, b] is a measure of how the function values change on that interval. Mathematically, it is defined as:

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

To calculate the average rate of change of a function on an interval, you need to know the function values at the endpoints of the interval, f(a) and f(b), as well as the difference between x-values, (b – a).

Let’s go through an example to illustrate how to calculate the average rate of change:

Example: Find the average rate of change of the function f(x) = 2x^2 – 3x + 1 on the interval [1, 4].

Step 1: Calculate the function values f(a) and f(b).
f(a) = 2(1)^2 – 3(1) + 1 = 2 – 3 + 1 = 0
f(b) = 2(4)^2 – 3(4) + 1 = 32 – 12 + 1 = 21

Step 2: Calculate the difference in x-values, (b – a).
(b – a) = 4 – 1 = 3

Step 3: Substitute the values into the formula for average rate of change.
Average Rate of Change = (f(b) – f(a)) / (b – a)
= (21 – 0) / 3
= 21 / 3
= 7

Hence, the average rate of change of f(x) on the interval [1, 4] is 7. This represents the average rate at which the function values change over that interval.

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