Three people set off on a car trip. They all start at the same time and end at the same time. Learn what calculus says about how fast they traveled along the way as you study the Mean Value Theorem in this lesson.
The Average Rate of Change of Three Drivers
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Have you ever noticed that when you get directions online, you’re usually given an estimate of how long it will take you to get there? For long trips, this estimate is often based on an average speed of about 62 miles per hour (mph). But not everybody drives 62 mph. Let’s look at the way three different people drive, and let’s plot their location as a function of time.So here I’ve got 62 mph for reference. Normal Nate is an average driver. Normal Nate tends to sit around 62 mph when he’s driving; sometimes he’s going 60, sometimes he’s going 65, but he’s pretty close to 62.
So if you look at his location as a function of time, he’s pretty close to the line where he would be if he were going exactly 62 mph. Constant Clara, on the other hand, drives exactly 62 mph the entire way. Late Leo doesn’t really go anywhere for a while, and then he speeds really fast to get to his destination.
Instantaneous Rate of Change
Understanding the Mean Value Theorem
The Mean Value Theorem says that for a function where you always have an instantaneous rate of change, the average rate of change will be equal to the instantaneous rate of change somewhere in the region where you’ve taken the average.So what does this mean? Let’s look at Normal Nate. Normal Nate averaged 62 mph for 2 hours. His average rate of change was 62.
We find that by taking the total number of miles he’s traveled (124) and dividing that by the amount of time that he spent (2). That’s his average rate of change. Now, at a couple of points on his travel path, he was going 62 mph exactly. At those points, the tangent to his position graph was equal to 62. So if this is the graph of his position as a function of time, he was going exactly 62 mph at three points between zero and 2 hours. At these three points, his instantaneous rate of change was equal to his average rate of change.Constant Clara was also going 62 mph, but she was going 62 mph exactly for her entire trip, so her instantaneous rate of change was equal to her average rate of change over the entire 2 hours.
Late Leo also averaged 62 mph, but he got a late start. If I look at his position as a function of time, I see that he was going 62 mph at the very beginning of his trip when he was starting to speed up and closer to the end of his trip when he was slowing back down. But he still hit 62 mph. His instantaneous rate of change was still 62 mph at some point along this 2-hour interval.
So let’s recap. The average rate of change over some interval from a to b, like 2 hours, is equal to your end point minus your start point divided by your region. So here we had your end point minus your start point, which was 124 miles divided by 2 hours. Our average rate of change was 62 mph. The instantaneous rate of change is the slope tangent to the curve at some point. It is exactly how fast you are going at any point in time.The Mean Value Theorem says that your average rate of change over some interval has to equal the instantaneous rate of change somewhere inside of that interval.
But that is only true if you always have a rate of change. So that’s only true if your velocity is always going to be defined.