If you’re anything like me then math wasn’t always your strongest subject in school. I remember countless hours spent staring at equations trying to make sense of them. But one equation that always baffled me was the average rate of change equation. What does it mean? And how do you calculate it?
Don’t worry I’m here to help! In this blog post I’m going to explain what the average rate of change equation means and how to calculate it. By the end of this post you’ll be a pro at calculating the average rate of change over an interval!
So what is the average rate of change equation?
Put simply the average rate of change equation is used to find the average rate of change of a function over a certain interval. In other words it allows you to find out how a function is changing on average over a certain period of time.
To calculate the average rate of change you need to take the difference between the function’s values at the endpoints of the interval and divide it by the length of the interval.
For example let’s say we have the function f(x) = x2. We want to know the average rate of change of this function over the interval from x = 1 to x = 3. To do this we first need to find the function’s values at x = 1 and x = 3.
f(1) = 1
f(3) = 9
Now we take the difference between these values and divide it by the length of the interval. The length of the interval is just the difference between the x-values so in this case it would be 3 – 1 = 2.
So the average rate of change would be:
(9 – 1) / (3 – 1) = 8 / 2 = 4
This means that on average the function f(x) = x2 is increasing at a rate of 4 over the interval from x = 1 to x = 3.
Now that you know how to calculate the average rate of change let’s look at a few examples.
Example 1
Find the average rate of change of the function f(x) = 2x + 1 over the interval from x = 0 to x = 2.
First we need to find the function’s values at x = 0 and x = 2.
f(0) = 1
f(2) = 5
Now we take the difference between these values and divide it by the length of the interval. The length of the interval is just the difference between the x-values so in this case it would be 2 – 0 = 2.
So the average rate of change would be:
(5 – 1) / (2 – 0) = 4 / 2 = 2
This means that on average the function f(x) = 2x + 1 is increasing at a rate of 2 over the interval from x = 0 to x = 2.
Example 2
Find the average rate of change of the function f(x) = x3 – 2x + 1 over the interval from x = -1 to x = 1.
First we need to find the function’s values at x = -1 and x = 1.
f(-1) = -2
f(1) = 2
Now we take the difference between these values and divide it by the length of the interval. The length of the interval is just the difference between the x-values so in this case it would be 1 – (-1) = 2.
So the average rate of change would be:
(2 – (-2)) / (1 – (-1)) = 4 / 2 = 2
This means that on average the function f(x) = x3 – 2x + 1 is increasing at a rate of 2 over the interval from x = -1 to x = 1.
I hope this blog post has helped you to understand what the average rate of change equation means and how to calculate it. If you have any questions feel free to leave a comment below.
What is the average rate of change of function over the interval from x = 1 to x = 4?
The average rate of change is the change in y divided by the change in x or (f(4)-f(1))/(4-1) = (4-1)/(4-1) = 3
What is the average rate of change of function over the interval from x = -3 to x = 2?
The average rate of change is the change in y divided by the change in x or (f(2)-f(-3))/(2-(-3)) = (2+3)/(2+3) = 5
What is the average rate of change of function over the interval from x = 5 to x = 9?
The average rate of change is the change in y divided by the change in x or (f(9)-f(5))/(9-5) = (9-5)/(9-5) = 4
What is the average rate of change of function over the interval from x = -1 to x = 3?
The average rate of change is the change in y divided by the change in x or (f(3)-f(-1))/(3-(-1)) = (3+1)/(3+1) = 2
What is the average rate of change of function over the interval from x = 4 to x = 8?
The average rate of change is the change in y divided by the change in x or (f(8)-f(4))/(8-4) = (8-4)/(8-4) = 4
What is the average rate of change of function over the interval from x = 0 to x = 4?
The average rate of change is the change in y divided by the change in x or (f(4)-f(0))/(4-0) = (4-0)/(4-0) = 4
What is the average rate of change of function over the interval from x = -5 to x = 0?
The average rate of change is the change in y divided by the change in x or (f(0)-f(-5))/(0-(-5)) = (0+5)/(0+5) = 5
What is the average rate of change of function over the interval from x = 3 to x = 7?
The average rate of change is the change in y divided by the change in x or (f(7)-f(3))/(7-3) = (7-3)/(7-3) = 4
What is the average rate of change of function over the interval from x = -2 to x = 2?
The average rate of change is the change in y divided by the change in x or (f(2)-f(-2))/(2-(-2)) = (2+2)/(2+2) = 4
What is the average rate of change of function over the interval from x = 5 to x = 9?
The average rate of change is the change in y divided by the change in x or (f(9)-f(5))/(9-5) = (9-5)/(9-5) = 4
What is the average rate of change of function over the interval from x = -4 to x = 0?
The average rate of change is the change in y divided by the change in x or (f(0)-f(-4))/(0-(-4)) = (0+4)/(0+4) = 4
What is the average rate of change of function over the interval from x = 1 to x = 5?
The average rate of change is the change in y divided by the change in x or (f(5)-f(1))/(5-1) = (5-1)/(5-1) = 4
What is the average rate of change of function over the interval from x = -3 to x = 1?
The average rate of change is the change in y divided by the change in x or (f(1)-f(-3))/(1-(-3)) = (1+3)/(1+3) = 4
What is the average rate of change of function over the interval from x = 4 to x = 8?
The average rate of change is the change in y divided by the change in x or (f(8)-f(4))/(8-4) = (8-4)/(8-4) = 4
What is the average rate of change of function over the interval from x = -1 to x = 3?
The average rate of change is the change in y divided by the change in x or (f(3)-f(-1))/(3-(-1)) = (3+1)/(3+1) = 2
