Calculating Average Rate of Change: Formula and Example 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] can be calculated using the formula:

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

To understand this formula, let’s break it down step-by-step:

1

The average rate of change of a function f(x) on the interval [a, b] can be calculated using the formula:

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

To understand this formula, let’s break it down step-by-step:

1. Calculate the difference in the values of f(x) at the endpoints of the interval: Subtract f(a) from f(b).

2. Find the difference in the input values (x-values) at the endpoints of the interval: Subtract a from b.

3. Divide the difference in function values by the difference in input values to get the average rate of change.

This formula measures the average rate at which the function f(x) changes over the interval [a, b]. It provides an overall rate of change over this interval rather than the instantaneous rate at any specific point.

Here is an example to illustrate how to calculate the average rate of change:

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

Step 1: Calculate the difference in function values:
f(b) – f(a) = f(4) – f(1)
= (2(4) + 3) – (2(1) + 3)
= (8 + 3) – (2 + 3)
= 11 – 5
= 6

Step 2: Calculate the difference in input values:
(b – a) = (4 – 1)
= 3

Step 3: Divide the difference in function values by the difference in input values:
Average rate of change = (f(b) – f(a)) / (b – a)
= 6 / 3
= 2

So, the average rate of change of f(x) = 2x + 3 on the interval [1, 4] is 2.

This means that on average, the function increases by 2 units for every 1 unit increase in x over the interval [1, 4].

More Answers: