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Average and Instantaneous Rates of Change

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  • 0:03 What Is a Rate of Change?
  • 1:11 Instantaneous vs Average Rates
  • 2:02 Calculating from a Graph
  • 3:10 Calculating with Calculus
  • 4:42 When the Rates Are the Same
  • 5:28 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

When you drive to the store, you're probably not going the same speed the entire time. Speed is an example of a rate of change. In this lesson, you'll learn about the difference between instantaneous and average rate of change and how to calculate both.

What Is a Rate of Change?

Imagine that you drive to a grocery store 10 miles away from your house, and it takes you 30 minutes to get there. That means that you traveled 10 miles in 1/2 hour, at an average speed of 20 miles per hour. (10 miles divided by 1/2 hour = 20 miles per hour). The speed of your car is a great example of a rate of change.

A rate of change tells you how quickly something is changing, such as the location of your car as you drive. You can also measure how quickly your hair grows, how much money your business makes each month, or how much water flows over a dam. All of these, and many more, can be represented by calculating the average rate of change of a quantity over a certain amount of time.

One easy way to calculate a rate of change is to make a graph of the quantity that is changing versus time. Then you can calculate the rate of change by finding the slope of the graph, like this one. The slope is found by dividing how much the y values change by how much the x values change. Let's look at a graph of position versus time and use that to determine the rate of change of position, more commonly known as speed.

average rate of change

Instantaneous vs Average Rate of Change

Let's go back a moment and think about that grocery store trip again. We calculated that your average speed for the entire trip was 20 miles per hour, but does that mean that you were traveling at exactly 20 miles per hour for the entire trip? What about when you were stopped at a red light or were stuck in traffic that wasn't moving? During those times you weren't moving at all, so your speed was zero.

When you measure a rate of change at a specific instant in time, this is called an instantaneous rate of change. An average rate of change tells you the average rate at which something was changing over a longer time period. While you were on your way to the grocery store, your speed was constantly changing. Sometimes you were moving faster than 20 miles per hour and sometimes slower. At each instant in time, your instantaneous rate of change would correspond to your speed at that exact moment.

Calculating from a Graph

So, we saw that you could calculate the average rate of change by calculating the slope of a line, but does that work for instantaneous rates of change as well? In fact, it does, although you have to think about slope a little differently than you may have before.

If you have a graph of your position vs. time that is NOT a straight line and you want to calculate your instantaneous speed, you can draw a line, known as a tangent line, that only touches the graph at one point. The slope of this tangent line will give you the instantaneous rate of change at exactly that point.


instantaneous rate of change


As you can see from the calculation on this graph, v equals 20 meters divided by 5 seconds minus 1.5 seconds, meaning 3.5 seconds, which equals 5.7 meters per second. How does that compare to the average rate of change? To determine your average speed over the whole trip, calculate the slope of a line drawn from the first point on the graph to the last point.


average rate of change from a graph


As you can see from the calculations on this graph, our average speed equals 49 meters divided by 7 seconds, which equals 7 meters per second. That is our average speed.

Calculating with Calculus

When we found your instantaneous speed using a graph, we drew a tangent line that only touched the graph at one point, and then calculated the slope of that line to find the rate of change. In calculus, the slope of a line tangent to a graph is called a derivative. So, if you have an equation that describes the position of the car, you can find the derivative, and this will give you a new equation for the speed of the car at every instant in time.

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