Introduction to average rate of change | Functions | Algebra I | Khan Academy

• So we have different definitions for d of t on the left and the right, and let's say that d is distance and t is time. So this is giving us our distance as a function of time. On the left, it's equal to 3t plus one, and you can see the graph of how distance is changing as a function of time. Here is a line, and just as a review from algebra, the rate of change of a line we refer to as the slope of a line.

We can figure it out; we can figure out, well, for any change in time, what is our change in distance? In this situation, if we're going from time equal one to time equal two, our change in time, delta t, is equal to one. What is our change in distance? We go from distance equal to four meters at time equals one, to distance equal to seven meters at time equal two. So our change in distance here is equal to three.

If we want to put our units, it's three meters for every one second in time. Our slope would be our change in our vertical divided by our change in our horizontal, which would be change in d, delta d over delta t, which is equal to three over one. We could just write that as three meters per second. You might recognize this as a rate; if you're thinking about your change in distance over change in time, this rate right over here is going to be your speed.

This is all a review of what you've seen before, and what's interesting about a line, or if we're talking about a linear function, is that your rate does not change at any point. The slope of this line between any two points is always going to be three. But what's interesting about this function on the right is that is not true; our rate of change is constantly changing, and we're going to study that in a lot more depth when we get to differential calculus.

Really, this video's a little bit of a foundational primer for that future state, where we learn about differential calculus. The thing to appreciate here is to think about the instantaneous rate of change someplace. So let's say right over there, if you ever think about the slope of a line that just barely touches this graph, it might look something like that, the slope of a tangent line.

Then right over here, it looks like it's a little bit steeper, and then over here, it looks like it's a little bit steeper. So it looks like your rate of change is increasing as t increases. As I mentioned, we will build the tools to later think about the instantaneous rate of change, but what we can start to think about is an average rate of change.

The average rate of change, and the way that we think about our average rate of change, is we use the same tools that we first learned in algebra. We think about slopes of secant lines. What is a secant line? Well, we talk about this in geometry; a secant is something that intersects a curve in two points. So let's say that there's a line that intersects at t equals zero and t equals one.

Let me draw that line; I'll draw it in orange. So this right over here is a secant line, and you could do the slope of the secant line as the average rate of change from t equals zero to t equals one. Well, what is that average rate of change going to be? The slope of our secant line is going to be our change in distance divided by our change in time, which is going to be equal to, well, our change in time is one second—one.

I'll put the units here: one second. What is our change in distance? At t equals zero, or d of zero is one, and d of one is two. So our distance has increased by one meter, meaning we've gone one meter in one second. We could say that our average rate of change over that first second from t equals zero to t equals one is one meter per second.

But let's think about what it is if we're going from t equals two to t equals three. Well, once again, we can look at this secant line and we can figure out its slope. The slope here, which you could also use the average rate of change from t equals two to t equals three, as I already mentioned, the rate of change seems to be constantly changing, but we can think about the average rate of change.

So that's going to be our change in distance over our change in time, which is going to be equal to when t is equal to two. Our distance is equal to five—so one, two, three, four, five; so that's five right over there. When t is equal to three, our distance is equal to ten—six, seven, eight, nine, ten; so that is ten right over there.

Our change in time, that's pretty straightforward; we've just gone forward one second, so that's one second. Our change in distance right over here, we go from five meters to ten meters, which is five meters. So this is equal to five meters per second. This makes it very clear that our average rate of change has changed from t equals zero to t equals one to t equals two to t equals three. Our average rate of change is higher on this second interval than on this first one.

As you can imagine, something very interesting to think about is what if you were to take the slope of the secant line with closer and closer points? Well, then you would get closer and closer to approximating that slope of the tangent line. That's actually what we will do when we get to calculus.