 Hi everyone. We're going to talk now of rates of change as determined from graphs. This part's great because we get to really look at some very real-life applications. So let's start with our first one. The graph that you see here depicts the outstanding gross national debt per capita in the United States from 1990 through 2014. So if you take a look at the graph and just take in what it's telling us on the horizontal axis, you'll see the years and it looks like they go in increments of one year at a time and on the vertical axis we have the national debt per capita in dollars going from about $10,000 up to $60,000. So the first thing we're going to consider is trying to find the average rate of change in that per capita debt between the years 1990 through 2014. So from the beginning to the very end. So remember that average rate of change is a slope calculation. So in this case the rate of change and since we're going to be estimating these values from our graph to be mathematically correct, we should use an approximately equal to symbol and remember it's going to be the change in y over change in x essentially. In this case our y values are the debt. So let's refer to that as capital D for debt and we need to do the debt in 2014 minus that debt in the year 1990. So that's our change in y and our x's of course are the years themselves. So the denominator is easy. It's the numerator that we'll have to try to estimate these values from the graph. So let's go back to the graph and see if we can do our best to estimate these. So let's start with 2014, which is all the way up here on the right. So if we trace that over and see about where it hits, now this is where you might estimate it to be slightly different than I do and that's fine as long as you're pretty close. When I looked at this the first time I thought it was a little bit over 55,000. So maybe it's 56,000. So let's maybe use that for the year 2014 and down here in the year 1990, well, it's not quite halfway at 15,000. I thought it was almost halfway halfway. So maybe 12,500 for that. And again as you're doing this and even on our own tests and quizzes and even on the AP exam, as long as you're close, that's fine. We don't expect everybody to get an estimate the exact same value from the graph, but as long as you're close, that's fine. It's almost like we're really going for the process rather than the actual answer in the end. So let's use those values then for our whys. So we would have, so we said the value at the year 2014, we thought to be about 56,000 and 12,500 for the year 1990. Of course, there's a 24-year difference in there. So if you number crunch that, go ahead and give it a try. You should get 1,812.5. Now let's talk about what this tells us because interpreting these rates of change are really an important key component to what we're doing here. So what this tells us is that the per capita debt, because that's what the problem is all about, is the per capita debt, is increasing, or you could say rising, and we know that because the 1,812.5 came out positive. So you could say either rising or increasing, and it would be, this is dollars remember, because our y-values are dollar amounts, so it would be $1,012.50 per year, and that really just applies for the years from 1990 and 2014. So when you're asked to interpret or explain what these rates of change mean, that's really what we're looking for. So let's now talk about instantaneous rate of change. So it's an instantaneous rate of change. Notice that those exact words are nowhere in the problem, but notice what you're asked for in one particular year. That's what makes it an instantaneous rate of change. So once again, our rate of change, and it is only going to be an estimation, of course, as noted in other videos, you are most welcome to use the two squiggly lines as opposed to the equal sign with the dot over it. For approximately equal to. So here is 1995 right here. At this point right here. All right, now there's a couple different ways you could do this. You could draw your own tangent line to the curve as best as you can and choose two points on that line that you have drawn and use those to calculate the slope. That's option one. Option two is to use that Delta neighborhood idea and what we could use are these points and values around the year 1995. Specifically in 1994 and 1996 you can think of that as a Delta value really of just one year. That's another option. So you can either draw your own tangent line, choose the two points that lie on that line you've drawn and use those points to draw to calculate your slope, or you can go with the points directly here from the graph on either side of the year 1995. So I'm going to do it that way only because in recording this it's sort of difficult for me to draw a tangent line. So I admit I'm going to take a little bit easier way out. So if we wanted to do that in order to calculate this rate of change, it's going to be the debt in the year 1996 minus the debt in 1994 over the difference between those two years. So I estimated in 1996, I thought that was pretty much just a little bit shy of 2000, so I call it like 19,500. Again, if you think it's 20,000, that's fine. And for 1994, I thought that was a little bit less, so maybe 18,000 for that one. And of course there's a two-year difference. So when you work that out, hopefully you get 750. So think again of what that tells us. That tells us specifically in the year 1995 the per capita debt is rising approximately $750 per year. So let's do another one similar to that. Maybe you can pause this video at this point and give this one a shot on your own and then we can compare answers. Notice it's another instantaneous rate of change because we're looking for the rate of change in one particular year in this year 2010, which is right here. So maybe we could do this similar to the other one and calculate it as the debt in 2011 minus the debt in 2009 over the difference. So why don't you go ahead and pause this, give it a shot, and then we can compare answers when we're done. So the way I estimated this for 2011, which is right up here, I thought that to be about, oh, that's this one here, 48,000 perhaps, and 2009, this one over here, about maybe 39,000, over of course a two-year difference. So when I did it, I came out with 4,500. Again, as I mentioned earlier, these are problems in which really we're going for the process that you use to arrive at your answer. Not so much what your actual answer is in the end. It's more the process and a correct interpretation of the answer you get. So maybe you got 4,500, like I did. Maybe you got something slightly different. Everybody should have gotten a positive answer, though, and hopefully somewhere in that 4,500 ballpark. So what this would tell us is that in the year 2010, the per capita debt is rising approximately $4,500 per year. Notice how different that is from the previous one. And if you take a look at the graph, notice what's happening at the year 2010. Notice how steep it is there. And if we go back to the other one, notice how much more flat it is. All right, that's what's accounting for that difference in the slope. So let's take a look at another real-life example. This one's kind of interesting. It's dealing with the number of mobile cell phones subscriptions in millions in Africa from the year 2000 to 2011. This data was and generated in the year 2012 and at that time Africa had more than 650 million mobile cell phone subscribers. And that was more than either the United States or the European Union. I thought that was kind of interesting. One reason I picked this for our example. So let's once again take a look at it. We're only concerned with that blue-green graph, which is the graph for the mobile phone subscribers. Okay? So you can see it's obviously increasing. That's pretty evident. And it looks like starting at the year 2000, up until about 2003, it's kind of flat, but then it seems to spike up pretty quickly and increase from there. And you can see here, here's the actual data for the year 2000 and 2011. One thing you'll want to take note of is that the hash marks are after the year. So this first hash mark you see here is the year 2000. The second one is 2001. Just wanted to note that for you. So the first thing we're going to calculate is finding the average rate of change in the number of mobile phone subscribers between the years 2000 and 2001. So right off the bat, I'm sure you're thinking it's going to be positive because obviously it increased over time. And in this case, we can use those actual values they gave us because they told us how many subscribers there were in each of those years. 16.5 million and 648.4 million. So our average rate of change, once again, it's a slope calculation, change in y over change in x. So I'm going to use s for subscribers. So we need the number of subscribers in the year 2011 minus the number of subscribers in the year 2000 over the difference in the years. So in 2011, they told us there were 648.4 million. Now we're going to leave off millions because we can just attach that at the end. And in the year 2000, there were 16.5 million users. So when you divide that out, you get 57.445. Now remember that's in millions. So what this tells us is that between the years 2000 and 2011, the number of mobile phone subscribers in Africa increased 57.445 million per year. So now let's turn to instantaneous rate of changes and finding the rate of change in the number of subscribers in one particular year. So we're going to consider the year 2004, which would be right about here on the graph. Now as I mentioned in the earlier scenario, you have a couple of options. You could draw your own tangent on to the curve, pick out two points that lie on that line you've drawn and calculate the slope between those points. That would be one way to get an answer. Your other option, again, is to use the Delta neighborhood idea. So we could choose perhaps the years 2003 and 2005, one year on either side. Again, giving us a Delta value of one year and we'll have to try to estimate the Y values, of course, from the graph and use that to get our answer. So I'm going to do it that way again because it's a little difficult for me to draw a tangent line in this application that I'm recording for you. So our rate of change, again, it's only going to be an approximation because we need to estimate these values and it's going to be the number of subscribers we said we'll use in 2005 minus the number of subscribers in 2003, so one year on either side of the target year. So if we take a look at the graph, so here's 2005, kind of like right there, I sort of thought that to be maybe a little bit lower than 200. I was thinking about maybe 190 perhaps. And for 2003, which is right about here, I thought that to be about 50 or so. Again, you might disagree with me, which is fine. So that gave me an answer of 70 in the end. Now just to give you an idea of how your answers might vary, I had also done this on paper by drawing a tangent line to the curve and when I did it that way, I got an answer of 50. Notice there is a big difference there. Again, we're going really for the process that you use to obtain your answer. So what this is telling us is that in the year 2004, the number of mobile phone subscribers was increasing in Africa, 70 million per year. Alright, don't forget that's not just 70, it's 70 million per year. So let's try one more and perhaps this is one on which you could pause this and compare your answer to the one I'm going to get. But we are going to do it the same way. So let's see, 2009 is right around here. So once again, maybe we can try to estimate our rate of change by using the values for 2008 and 2010. So we're going to get the number of subscribers in 2010 minus the number we estimate for 2008. Alright, so why don't you pause this for a second, figure out what answer you get and then we'll compare with each other. So when I did this, I was estimating your 2010, which is up here. I thought that was about a little bit more than 600, so maybe like 610 and then for 2008 I thought that was about 400 or so. So the answer I got when I did it was 105. You can see how close you are to what I got. So again, think about what that's telling us. Since it's positive, it tells us the rate is increasing. So we would have in the year 2009 the number of mobile phone subscribers in Africa was increasing approximately 105 million per year. Now again, this is another one I had tried by drawing on paper a tangent to the curve and I got 100 that way. So pretty close to 105. So you can see how you're in certain pairs.