Calculating & Interpreting a Function's Average Rate of Change

Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught history, and has an MA in Islamic law/finance. He has since founded his own financial advice firm, Newton Analytical.

The average rate of change in a given function is helpful to identify speeds and other changing variables. Learn why these are used, and how they are calculated and interpreted through examples. Updated: 11/09/2021

What Is an Average Rate of Change?

So you're planning a trip from Boston to New York. You know that the distance is around 215 miles. Still, you're curious as to what speed you'll have to go in order to get there in 4 hours. It turns out that you've got a birthday party to make it to. Now, as important as that birthday party may be, you won't be going the same speed the whole time. After all, it's doubtful that you'll be leaving your driveway at 60 miles per hour.

Although your instantaneous speed, your speed at any given instant, may be important to state troopers looking to hand out high-priced driving tickets, it is not really that important here. Instead, we are looking for your average miles per hour over a multi-hour trip. An average speed is an example of an average rate of change, which is the difference of change of a quantity over the time it is changing.

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Why Use an Average Rate of Change?

Average rates of change are important for more than just trip planning. Let's say you were trying to fill up a new swimming pool. Wouldn't knowing the average rate at which your water hose can supply water help you figure out how long it would take you to fill your pool to fill up? In this example, the rate at which the hose supplies water is an example of a rate of change.

Or perhaps you're planning on lifting weights to gain muscle. Very often, you'll hear medical professionals advise against gaining more than a pound or so of muscle a week, on average. That extra pound may come as a half a pound one week or even two pounds the next. However, as long as it all averages out to around a pound a week over a period of time, many medical professionals won't be too alarmed.

How to Calculate Average Rate of Change

Luckily, being able to find an average rate of change is a pretty easy thing to do. Simply put, you just find the difference between the quantities at two different points, then divide that by how much time elapsed between those two points.

Sometimes when you're doing this, you'll have a negative answer. While that may seem scary, it's really not, not if you know what you're looking for. A negative rate of change means that the quantity is shrinking over time. Therefore, if you were losing weight, a negative rate of change would be a good thing. Meanwhile, if you were trying to fill up your swimming pool, a negative rate of change could indicate a disaster!

Examples of Average Rate of Change

Let's do a couple of problems on rate of change to make sure that you fully understand the concepts involved. For the first one, we'll do a rate of change that involves a quantity increasing. For the second one, we'll do one that shows a quantity decreasing. Let's begin.

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