Understanding Average Rate of Change

By the end of this video, you're going to be flying through these average rate of change problems like they're nothing. And in about 30 seconds, here's how this video is going to get you there. We're going to start off with this problem where we find the average rate of change of this function on some interval using the average rate of change formula. And after we do that problem, we're going to talk about what the average rate of change actually means graphically. After we do all that, We're going to move on to a harder average rate of change problem where the function gets a little more gross and the interval is also going to be in a different notation. So we'll talk about how to go through that. And after we do all that, I'll give you a problem to try and answer in the comments. And if you're looking for the notes for this video, and guys, these aren't just any old notes. These are printable notes with a QR code attached that will take you back to the video. And these notes have timestamps for the problems that we do in the video. These notes, they have it all. And if you want the link to those notes, that's going to be right in the description. So we're going to start off with this first problem here, where we have to determine the average rate of change of f of x, which is x squared plus 3, from x equals negative 1 to x equals 3. Now, to find the average rate of change, we're going to use the average rate of change formula, which is right over here on the right. And that formula has four different pieces. It's got f of b, f of a, and it's also got b and a. So let's start off by talking about that b and the a. Now, a and b, that is our interval. Those are our x values that we're given. in the problem. The first x value that we're given is a and the second x value that we're given is b and right there now that we have that we have the first two pieces of the formula out of the way. Now we just need to figure out what f of a and f of b are. So to do that we're going to need our function f of x which we know in the problem says that it's x squared plus 3 and then if we want to find f of b I'll find that one first because it's first in the formula b is 3 So f of b can be rewritten as f of 3. Just plug in the 3 for b. And then how do we find the value of f of 3? Well, let's bring over that f of x. The only difference between f of x and f of 3 is that we have a 3 plugged in for x. And so all we're going to do with this x squared plus 3 is we're going to plug a 3 in for x. So instead of being x squared, it's going to be 3 squared. And then we'll add 3. Now 3 squared. That's 3 times 3. That's 9. So we have 9 plus 3, and that's 12. And right there, we have f of b. Now, what about f of a? Well, a is negative 1 here. So we can rewrite this as f of negative 1. Just plug in that negative 1 for a. And then we just plug in a negative 1 everywhere we see an x in x squared plus 3. So we're going to get, instead of x squared, it's going to be negative 1 squared. and then we add our 3. Now negative 1 squared that's negative 1 times negative 1 that is a positive 1 and so we get a positive 1 plus 3 and that's 4. And now we have everything that we need to find the average rate of change. So we want to do f of b minus f of a all over b minus a and we know that f of b this first piece we found that to be 12. And we found f of a to be 4. So we're going to have 12 minus 4. And then we have that b is 3, and it's minus a. What is a? a is negative 1. And so we're going to get negative 1 here. Now, this is something that I want to draw your attention to because a lot of people will mess up here. We have two negatives here, not just one. And let me show you where each of those negatives come from. The first negative that you see here comes from the fact that we have b minus a. The other negative is part of a itself a has a negative on it that's where the two negatives come from there's one here and there's another one being a part of a so you're not just going to only put one negative there there's two of them and that matters a lot for your answer 12 minus 4 that's 8 and then we have 3 minus negative 1 we see two negatives you can make them both positive that's going to be 3 plus 1 which is 4 and 8 divided by 4 is 2 And that is our answer for problem one. So that is how you find the average rate of change. But you might be wondering at this point, OK, well, what actually is the average rate of change? What does that mean graphically? So let's talk about that. Here I have the curve that we're talking about in red. That's our function. And I have our x and y-axis here. What the average rate of change is saying is we're going to be, let's pick two x values here. We'll pick a and b. And I'll draw lines up. So we can get to the points on the curve at a and b. Now, the average rate of change is going to give us the slope of this secant line that I've just drawn here. That's secant spelled like this. And that is not to be confused with a tangent line, which a tangent line. It'll only touch the curve at like one point. So a tangent line will look something like this. It'll only touch the curve once. That's what a tangent line is. A secant line is going to go from a point to another point on the curve. And the average rate of change is going to be the slope of this line. And let me show you why that actually means the average rate of change along this curve. We can pick a few points on this line where the slope is farther down. It's more downward than the slope of the secant line. And there's other points on the curve where the slope is going more upwards. than the slope of the secant line. And all these points along the curve, all the slopes at these different points, they're going to average out to be the slope of the secant line. And so that's what the average rate of change actually tells you. It tells you the average direction that you're traveling as you go along this curve from A to B. And that is given by the slope of the secant line. That's how you can find it. And that's where the average rate of change formula actually comes from. So with all that being said, let's move on to our harder problem here. Problem two, which has to determine the average rate of change of f of x, which is now x cubed minus 2x squared plus 3 on the interval from negative 2 to 2. Now this is definitely a different looking interval than what we were given to start off with. We were given x equals negative 1 to x equals 3 and it was written a lot differently. Now our x values are given in this bracket notation and that's fine. Literally this just means from x equals negative 2 to x equals 2. That's how you read that. And so the first number is going to be your a, and the second number is going to be your b. And now that we found those two, all we need to find is f of b and f of a. Now the function in question is x cubed minus 2x squared plus 3, and if we want to find f of b, we know that b is 2 here, so we plug 2 in for b. And now again, how do we find f of 2? Well everywhere we see an x over here, we're going to be putting a 2. So this is going to be 2 cubed instead of x cubed minus 2 times 2 squared plus 3. And that is going to take a little bit of work to expand out, but not too bad. 2 cubed is 2 times 2 times 2, which is 8. And that's going to be 8 minus a 2 times, what's 2 squared? 2 squared is 4, so this is a 2 times 4, which is 8. So this will be minus 8. and then we have that plus 3 on the end. Now 8 minus 8, that's 0. 0 plus 3 is 3. And now we can move on to f of a. f of a, a is negative 2 here. We plug in a negative 2 everywhere we see an x, and that'll give us a negative 2 cubed minus 2 times negative 2 squared plus 3. Now negative 2 cubed, well what is that? Let's think about it. Negative 2 times negative 2 times negative 2. This right here, negative 2 times negative 2, that's 4. 4 times negative 2 is negative 8. And then we have a negative 2 squared here, that's a positive 4. Positive 4 times 2 is 8. And remember there's a minus sign here, so this is going to be minus 8. And then we have plus 3 on the end, so this is going to be a negative 16 plus 3, which is negative 13. And now we have everything we need to find the average rate of change. We know that f of b, that was f of 2, we found that was 3. And we know that f of a is negative 13. And remember, there is two negatives here. Because it's minus f of a, and, well, f of a... is negative. That's a negative 13 there. The same thing is actually going to happen with b minus a because a is negative. So b is 2 and a is negative 2. So b, plug that 2 in, and then we have minus a, which is negative 2. And now if we have two negatives next to each other, they both become positive. So we're going to get two positives right here. And we also get two positives right here. So we have a 3 plus 13, that's 16. And in the denominator, we have 2 plus 2, that's a 4. So we have 16. divided by four and that is four and that's our answer for the last problem for this video so hopefully that gets you feeling a lot better with these average rate of change problems and if you do feel pretty good with that then here is a problem for you to try and and answer in the comments. This problem is asking you to determine the average rate of change of this function here on the interval from negative three to zero. And I give you the formula again for the average rate of change. So give that problem a shot. Let me know what your answer is in the comments. And if you have any questions on anything we talked about in this video, again, let me know in the comments and I'll try to get back to you when I can. Now remember the notes for this video are linked right in the description. I know you're gonna want the notes. You might as well just snag them. They're free. And yeah, make sure that you're subscribed to the channel because look, we. We each have our goals. We're trying to get there. You're trying to pass math. I'm trying to not be poor. We can help each other. I just helped you. You can help me by liking the video and subscribing to the channel, all right? We gotta, we're gonna hit 100K eventually. Anyways, that's gonna do it for this video, guys. And I will see you soon.

So we'll talk about how to go through that. And after we do all that, I'll give you a problem to try and answer in the comments. And if you're looking for the notes for this video, and guys, these aren't just any old notes.

These are printable notes with a QR code attached that will take you back to the video. And these notes have timestamps for the problems that we do in the video. These notes, they have it all.

And if you want the link to those notes, that's going to be right in the description. So we're going to start off with this first problem here, where we have to determine the average rate of change of f of x, which is x squared plus 3, from x equals negative 1 to x equals 3. Now, to find the average rate of change, we're going to use the average rate of change formula, which is right over here on the right. And that formula has four different pieces. It's got f of b, f of a, and it's also got b and a.

So let's start off by talking about that b and the a. Now, a and b, that is our interval. Those are our x values that we're given.

in the problem. The first x value that we're given is a and the second x value that we're given is b and right there now that we have that we have the first two pieces of the formula out of the way. Now we just need to figure out what f of a and f of b are. So to do that we're going to need our function f of x which we know in the problem says that it's x squared plus 3 and then if we want to find f of b I'll find that one first because it's first in the formula b is 3 So f of b can be rewritten as f of 3. Just plug in the 3 for b.

And then how do we find the value of f of 3? Well, let's bring over that f of x. The only difference between f of x and f of 3 is that we have a 3 plugged in for x.

And so all we're going to do with this x squared plus 3 is we're going to plug a 3 in for x. So instead of being x squared, it's going to be 3 squared. And then we'll add 3. Now 3 squared.

That's 3 times 3. That's 9. So we have 9 plus 3, and that's 12. And right there, we have f of b. Now, what about f of a? Well, a is negative 1 here.

So we can rewrite this as f of negative 1. Just plug in that negative 1 for a. And then we just plug in a negative 1 everywhere we see an x in x squared plus 3. So we're going to get, instead of x squared, it's going to be negative 1 squared. and then we add our 3. Now negative 1 squared that's negative 1 times negative 1 that is a positive 1 and so we get a positive 1 plus 3 and that's 4. And now we have everything that we need to find the average rate of change. So we want to do f of b minus f of a all over b minus a and we know that f of b this first piece we found that to be 12. And we found f of a to be 4. So we're going to have 12 minus 4. And then we have that b is 3, and it's minus a.

What is a? a is negative 1. And so we're going to get negative 1 here. Now, this is something that I want to draw your attention to because a lot of people will mess up here.

We have two negatives here, not just one. And let me show you where each of those negatives come from. The first negative that you see here comes from the fact that we have b minus a. The other negative is part of a itself a has a negative on it that's where the two negatives come from there's one here and there's another one being a part of a so you're not just going to only put one negative there there's two of them and that matters a lot for your answer 12 minus 4 that's 8 and then we have 3 minus negative 1 we see two negatives you can make them both positive that's going to be 3 plus 1 which is 4 and 8 divided by 4 is 2 And that is our answer for problem one. So that is how you find the average rate of change.

But you might be wondering at this point, OK, well, what actually is the average rate of change? What does that mean graphically? So let's talk about that.

Here I have the curve that we're talking about in red. That's our function. And I have our x and y-axis here.

What the average rate of change is saying is we're going to be, let's pick two x values here. We'll pick a and b. And I'll draw lines up. So we can get to the points on the curve at a and b. Now, the average rate of change is going to give us the slope of this secant line that I've just drawn here.

That's secant spelled like this. And that is not to be confused with a tangent line, which a tangent line. It'll only touch the curve at like one point.

So a tangent line will look something like this. It'll only touch the curve once. That's what a tangent line is. A secant line is going to go from a point to another point on the curve.

And the average rate of change is going to be the slope of this line. And let me show you why that actually means the average rate of change along this curve. We can pick a few points on this line where the slope is farther down. It's more downward than the slope of the secant line.

And there's other points on the curve where the slope is going more upwards. than the slope of the secant line. And all these points along the curve, all the slopes at these different points, they're going to average out to be the slope of the secant line.

And so that's what the average rate of change actually tells you. It tells you the average direction that you're traveling as you go along this curve from A to B. And that is given by the slope of the secant line.

That's how you can find it. And that's where the average rate of change formula actually comes from. So with all that being said, let's move on to our harder problem here. Problem two, which has to determine the average rate of change of f of x, which is now x cubed minus 2x squared plus 3 on the interval from negative 2 to 2. Now this is definitely a different looking interval than what we were given to start off with. We were given x equals negative 1 to x equals 3 and it was written a lot differently.

Now our x values are given in this bracket notation and that's fine. Literally this just means from x equals negative 2 to x equals 2. That's how you read that. And so the first number is going to be your a, and the second number is going to be your b. And now that we found those two, all we need to find is f of b and f of a.

Now the function in question is x cubed minus 2x squared plus 3, and if we want to find f of b, we know that b is 2 here, so we plug 2 in for b. And now again, how do we find f of 2? Well everywhere we see an x over here, we're going to be putting a 2. So this is going to be 2 cubed instead of x cubed minus 2 times 2 squared plus 3. And that is going to take a little bit of work to expand out, but not too bad. 2 cubed is 2 times 2 times 2, which is 8. And that's going to be 8 minus a 2 times, what's 2 squared?

2 squared is 4, so this is a 2 times 4, which is 8. So this will be minus 8. and then we have that plus 3 on the end. Now 8 minus 8, that's 0. 0 plus 3 is 3. And now we can move on to f of a. f of a, a is negative 2 here.

We plug in a negative 2 everywhere we see an x, and that'll give us a negative 2 cubed minus 2 times negative 2 squared plus 3. Now negative 2 cubed, well what is that? Let's think about it. Negative 2 times negative 2 times negative 2. This right here, negative 2 times negative 2, that's 4. 4 times negative 2 is negative 8. And then we have a negative 2 squared here, that's a positive 4. Positive 4 times 2 is 8. And remember there's a minus sign here, so this is going to be minus 8. And then we have plus 3 on the end, so this is going to be a negative 16 plus 3, which is negative 13. And now we have everything we need to find the average rate of change.

We know that f of b, that was f of 2, we found that was 3. And we know that f of a is negative 13. And remember, there is two negatives here. Because it's minus f of a, and, well, f of a... is negative.

That's a negative 13 there. The same thing is actually going to happen with b minus a because a is negative. So b is 2 and a is negative 2. So b, plug that 2 in, and then we have minus a, which is negative 2. And now if we have two negatives next to each other, they both become positive. So we're going to get two positives right here.

And we also get two positives right here. So we have a 3 plus 13, that's 16. And in the denominator, we have 2 plus 2, that's a 4. So we have 16. divided by four and that is four and that's our answer for the last problem for this video so hopefully that gets you feeling a lot better with these average rate of change problems and if you do feel pretty good with that then here is a problem for you to try and and answer in the comments. This problem is asking you to determine the average rate of change of this function here on the interval from negative three to zero. And I give you the formula again for the average rate of change. So give that problem a shot.

Let me know what your answer is in the comments. And if you have any questions on anything we talked about in this video, again, let me know in the comments and I'll try to get back to you when I can. Now remember the notes for this video are linked right in the description. I know you're gonna want the notes. You might as well just snag them.

They're free. And yeah, make sure that you're subscribed to the channel because look, we. We each have our goals. We're trying to get there.

You're trying to pass math. I'm trying to not be poor. We can help each other.

I just helped you. You can help me by liking the video and subscribing to the channel, all right? We gotta, we're gonna hit 100K eventually.

Anyways, that's gonna do it for this video, guys. And I will see you soon.

Transcript for:Understanding Average Rate of Change

Transcript for:
Understanding Average Rate of Change