 This video is going to talk about exponential logarithmic and logistical equation models. And we can also model linear and quadratic, we've done a lot of that, but we do need to know what they look like so that what we're going to end up doing is looking at some scattergrams and it will tell us what kind of model we need to do as well as what data we have. So let's talk about linear. In linear, the y changes the same amount each time. Okay, so you have something like, if I had a table, I'd have something like 1, 2, 3 and y would be 5, 10, 15. Consistent change where we're adding or subtracting. And the graph is going to look something like this if it's increasing and it'll look like this if it's decreasing. So if I have something that looks kind of like that on my scattergram, I would think linear. Well, quadratics, they both increase and decrease. They're characteristic because remember the graph is going to look something like this. So this particular graph decreases and then it increases at the end. Or if it were an upside down parabola, it would increase and then get to the vertex and decrease. In a table, it would look something like 0, 1, 2, 3, and we might see something like 5, 9, 5, 4. So we can see that it increased but then it decreased. Exponential is going to have a grower decay. Use the working models so it's the growth or decay, the kinds of problems that they are often a population problem. Population of bacteria, population of people, population of atoms, all kinds of things. And if you remember, the graphs look something like this. So they're either going to be increasing or decreasing. This is a growth and this is a decay. If you think about your table, increases, this one starts out slow and increases rapidly. This one decreases rapidly and then kind of levels out a little bit. So if I again think of my 0, 1, 2, 3, I might have something like 5 and then I would have something like 20 and then maybe 80 and then 250. So it's growing rather quickly. And then finally we have our logarithmics and we haven't necessarily looked at these graphs very much. But remember they're inverses of exponentials. If I put my x and y axes in here, this would be my exponential and this could also be my exponential. They hug the x-axis but they go through the y. So logarithms remember are the inverse of that. We're going to have a graph that looks something like, let me get my axes up here first. And they're going to hug the y-axis and they can come up through here, they can come up through the x-axis or they're going to decrease rapidly and then kind of level off. So I will rapidly and this levels up over here, decreases rapidly but then kind of levels off but it doesn't have any negative x's. Okay, so the domain is x is greater than 0. Over here we would have said that y was greater than 0. That was the important part over here. So let's see what we can do. We have this data here and we want to graph the data and then explain why an exponential model would be best. So go ahead and take a little bit of time to put that in there and in a moment I will show you my scattergram and I get this scattergram. Okay, that's not perfect but it looks very much like an exponential curve. If I kind of went like this, something like that, I mean that's a good guess. But it looks exponential. So one reason it might be exponential is because of my graph. From 0 to 15 it didn't really take very much but I went from 2.5 up to 30. That's a lot bigger. So it grew kind of slowly at the beginning. That's what was happening in here. It's kind of growing slowly in here but then it started taking off. Okay, so here we have accumulated weighted diamonds extracted from a diamond mind. So we have these months and these curate weights. So go ahead and put those in there and graph it and then we'll talk about why it should be a logarithmic model. So here's my graph and if you look at it, it looks like I might be able to fit in a logarithmic. Okay, it increases rather quickly but then it kind of starts to level off. It's not exponential because it's going the wrong direction. It's curving the wrong way for an exponential. And why might it be logarithmic? Well, if you think about this, X has to be greater than zero, although months are going to be greater than zero. But you can see that we started out kind of a big gap in here but then they're getting closer and closer and closer. So that's like the opposite of what happens with an exponential growth. This is a logarithmic growth but it starts out big and then in slow. All right, moving along, here's the new one. Logistic growth model. At first growth is very rapid but it begins to taper off and slow down as the environmental factors are used up. So given constants A, B and C, the logistic growth population depends on time according to this model. And C here is what we call the carrying capacity of the population. It's, that is T gets really large, it tends towards C and we're going to watch that in our calculator. So we have one plus A times E to the negative BT is our function and that's going to always be the way a logistic model sets up. So that's something we need to keep in mind. Okay, so our population is going to get really close but never quite get to C. So let's look at a problem. And let's go ahead and take time and put this in your calculator. So here's my scattergram. And you can see that this one kind of has S looking kind of shape. So it kind of starts out a little, growing a little quickly and then it grows really rapidly but it starts to taper off. And it looks like it almost flattens out even more than that logarithmic. So that's what we can call our logistic looking model. But the context of the model, we're talking about percentage of cable TVs. Well, eventually you're going to saturate the market, so it's going to level off. That's why we would consider this to be a logistic. Let's go and move along and see. It says, attempt to estimate the carrying capacity. Well, it looks like we'd have to look at our table a little bit better. This is going to be 70, this dot here is that's 2870. So it looks like it might be about maybe 70, we can try, we can guess. We're just estimating, it says attempt to estimate. So we're going to say it's approximately 70, might be a little bit more. And then it says find the regression model. So here we go, here's what we really need. I've got all this data in here, just like you do. We go back to stat, over to calculate if you remember how to do regressions. But if you arrow down way past here, we're going to get to B. So C where B says logistic, so we're going to do stats, over to calculate. And then we want to choose B, which says logistic. You can also find that, just for showing you that. If I'm over here, I could also find that by alpha. And then the B is above your apps, and it would take you right to logistic. If you don't want to have to arrow down all the time. So this is alpha and then apps. Okay, back to my calculator, see if I can stick with it this time. So I'm logistic, and then I just press enter, because that's the one I want. And this one takes a little while, I've noticed. But you can see this little thing going here, so it's thinking. And now we have it, okay? So it tells you the model there. So Y is equal to, and then if you have yours up, I have mine on my paper, so I can remember. Y is equal to, it was 66.9, double check that. Beware, this formula starts with C, but A is the first thing you see. So you need to come down here to C and say that that's 69.99, I'll call it. Divided by 1 plus A, which we'll call this 3.9998 or 998, we will call it 4. E to the negative B, so negative 0.22x. So again, what did we just say? Y is equal to 69.99, that's C. Over 1 plus A, which was 3.998, so we'll round it to 4. Times E to the negative 0.22 times T. So what, if we look at this then, remember C was 69.99, and we said approximately 70, so we were right on. What is the percentage of households with cable TV in 1999? Well, if you take 1999 and subtract when it began, which was 1976, we find out that there's 23 years. Now I've called up my calculator. I've put in this function, so there's my 69.99 divided by, and then I had to put the denominator in parentheses, over 1 plus 4 times E, and then my calculator, I had to put parentheses around my exponent, and then I had to close the denominator, that's the way I put it in there. And then I want to do second window, because I know I want X to be 23. And if I go to look at my table then, I find out that it's going to be 69.99. I've hit the carrying capacity for this thing. But another thing that I'd like to show you is the graph. And I should show you my window. I've looked at, I've played with this so that it will look like what we want to see. I'm going from negative 2 to 2, because this grows really fast. And then I'm going to be 0.1, so I can see things that are happening. And I went all the way up to 75, because I knew that it was going to even out at about 70. So if I look at my graph, you can see that it definitely has that logistic curve. And this grows really fast at the beginning, and it levels off right away. Alright, let's try one more. Longer an area is mined for gold. The more difficult and expensive it is, it will retain cumulative total advances produced by a particular mind as shown in the table. So draw a context from a scatter plot, and then use it in the context to determine the best regression model to find the equation. So if you put all that data in your calculator, pause and put that in your calculator, you find out that you get this graph. And if you look at this graph, it looks very, can you guess before I tell you? The graph looks logarithmic. And if we look at what's happening here, we grow rather rapidly, but then it kind of starts tapering off, so that could be very well be logarithmic. But the scatogram is usually the real telltale sign. So if we continue on then, what is the total of number of ounces mined? Well, we're going to have to go back and find our regression. And we might as well, oh, I'll write it on that page since we know. So our model says in here, I'll write it down here by this last question. We are going to do, you're going to need your calculator again. And I need to show you, because it's something new. All right, so we have, go into our stats, this data. And I need to go to stat and over to calculate. This is how I do all my regressions. But now, again, we're going to arrow this time, but you won't have to every time. I'll tell you what it is. And go one more to nine. And you see Len Reg. Okay, remember, it doesn't matter if it's a log or a natural log. So we're going to do Len LN Reg. Let's call it LN Reg. So press enter and then enter again, because that's what we want. And we get this as A times B LNX, right? So we have A and B, write those down. And I will put it in here for you. Y is equal to negative 2,635.58 plus my B was 1904.83 and then LNX. So using that formula, what was the total number of ounces mined after 18 months? X is months. So we're going to have Y is equal to, I'm going to need our calculator here, 2,635.58, you can put that in your calculator, plus 1904.83 and then LN18, because I know what that one is. And when we put that in there, we should find out that we have approximately 2,870 ounces. So now it says, how about how many months did it take to mine a total of 4,000 ounces? Well, 4,000 then is going to be our Y. So we have 4,000 equal to this negative 2,635.8 plus our 1904.83 LNX. I have to add the 2,635.58, so that'll give me 6,635.58 and that's equal to the 1,904.83 LNX. And I want LNX completely by itself now, so I'm going to divide off. So 6,6.58 divided by 1,904.83 and that's equal to LNX and now I'm ready to exponentiate. So I can say E to this side and E to the LNX. And remember when you exponentiate the base of E and the LN cancels each other out. So I just have X is equal to E to this thing. Well, that's not going to be good enough. We'll come over here and say it's approximately, I've already put in all the numbers. I just have to press enter to find out what I have here. And it says 32.575, so we'll say it's close enough to say that it's approximately 33 and we're in months. Let's do our very last problem, says according to this model what is projected total for after 50 months. And this one we don't know, but we're just going to come in here and let X be 50 because it was just a plug and chug. And we find out that this is 4,816.16 ounces.