 Welcome back everyone. In this video, we're going to talk about the average rate of change of a function. So imagine we have a function f of x which is defined on some closed interval a to b. a and b are both included inside the domain of that. We can talk about the average rate of change of the function with respect to that interval a to b. So with respect to the variable x on that interval a to b, it's the following formula. We call this delta y over delta x and we often indicate the interval a to b because the average rate of change does depend on the interval. If the interval is clear, we might drop it out there. But the average rate of change delta x over delta y over delta x, excuse me, this is going to equal the formula f of b minus f of a over b minus a. Now I want to point out to you that this right here is just the slope formula, the average rate of change formula that we've seen already. But why this is really important is going to be the following basic idea. If we have a function, let's say, you know, not a line, our function is going to be curvy, so maybe it does something like this. If we pick two points in the domain a and b right here, we take this point right here, we take this point right here. What if we could forecast the the function as just a linear? What if we just pretend for a moment it was a linear function? What if we just connected the dots, right? We play a little bit of kindergarten math right there and we just extend the line like that. We get this linear function that just goes between a and b. So as the function transitions from a to b, on average it's really not any different than the line that we see right here. And so this line is commonly referred to as a secant line. This is based off of trigonometry because if you have a circle, a secant line is a line that intersects the circle of two different places. So we're kind of playing along that analogy right there. A secant line is going to be a line that passes through a function at two distinct points. Now the slope of that secant line will just be delta y over delta x on the interval a to b. So the average rate of change is the slope of the secant line. If we could approximate the function using a linear estimate, the slope of that line would be the average rate of change. The average rate of change is measuring the steepness of this line, the steeper the line, the faster the function's changing. The less steep the line is, the slower the function's changing. So let's do some calculations with this regard, right? So f of x equals 3x squared. Can we find the average rate of change on the interval of 1 to 3? And so if we want to do that, we do delta y over delta x as you go from 1 to 3. This just means we're going to take f of 3 minus f of 1 over 3 minus 1. The denominator is easy enough. You're going to get a 2. If we plug 3 into our function right here, you're going to get 3 squared, which is 9, times that by 3, you get 27. And if you subtract from that f of 1, if you plug in x into there, you get 1 squared, which is 1, times 3, which is 3. So you get 27 minus 3. That's going to be 24 over 2. That would divide to give us 12. So 12 is the average rate of change of the function from 1 to 3. So on average, you are increasing, you're going to increase 12 units along the y-scale every time you increase the x by 1. If the function were a line, its slope would be 12. What about the average rate of change from 1 to 4? Or 1 to 5, excuse me. In that case, delta y over delta x as you go from 1 to 5 here, this would look like f of 5 minus f of 1 over 5 minus 1. 5 minus 1 is going to be 4. f of 1, we already did that one. That's 1. f of 5 this time, you would take 5 squared, which is 25, times that by 3, you get 75. Oh, I'm sorry, f of 1 is not 1. 1 squared is 1. But then you times that by 3, you get 3. So f of 1 was 3. My bad there. 75 minus 3 is 72 divided by 4. And that turns out to be 18. So on average, the rate of change as you go from 1 to 5 would be 18 y's per x. And then if we do this one more time, the average rate of change from 1 to 7, the square root of y over, sorry, not the square root, just delta y over delta x as you go from 1 to 7 right here. Same formula. You're going to get f of 7 minus f of 1 over 7 minus 1. 7 minus 1 is 6. f of 1 is 3. And notice I got to write that time sweet. And if you plug 7 inside of the function, you're going to get 7 squared, which is 49, times that by 3, you get 147. Taking away the 3, you're going to get 144 over 6, which then simplifies to be 24. So notice as you go from 1 to 7, the slope of that secant line is going to be 24. So on average, every time you increase one horizontal unit, you'll increase 24 vertical units. And so I want to I want you to compare these things all together, right? As you went from 1 to 3, the average rate was 12. As you go from 1 to 5, the average rate was 18. As you go from 1 to 7, the average rate was 24. Notice that the average rate of change as you go farther and farther from 1 is getting bigger. We'll talk about that in just a second. Look at these pictures right here. In the yellow, you see the parabola y equals 3x squared. For the sake of scale, notice the x-axis. This right here is two units, but this right here is 40 units. So there is a distortion on the y-scale right there. But our parabola is this yellow curve right here. These green segments are the secant lines. The slope of the first secant line right here would be 12. The slope of the next one would be 18. The slope of the next one would be 24. Notice that the farther down the line I go, the steeper the secant line is getting. And that is really just me saying that the graph is concave upward, right? So what we see here is that if your average rate of change is increasing, right? So if you have a fixed point at x equals 1, if you see that the average rate of change gets bigger and bigger and bigger the farther away you get from your fixed point, that means your function is concave upward. On the other hand, if you had a concave down graph and you took some fixed point, you're going to see that your secant lines are going to get, their slopes going to get lower and lower and lower because it's more negative each and every time. That would indicate that your graph is concave downward. So this idea of concavity is directly related to average rates of change. If your rates of change are going up, you're concaving up. If your rates of change are going down, that means you're concaving down. And the two things are connected to each other. And it's a pretty big deal, right? The rate of change is basically telling you how quickly is our function increasing or how quickly is it decreasing. If you compare two different increasing functions, if you compare their average rates of change for some fixed point, the one who has a faster growing rate of change would indicate it's increasing faster than, say, the other one. So these rates of change have important applications to studying functions, which we'll see some more of these in the future, of course.