 This lesson is on slope fields. Slope fields are graphs of functions and relations created from their derivatives. We think of slopes and skiing, and maybe that's a good way to think of graphing slope fields because they're undulating normally and there are several of them and several levels of them. So let's go on and do something with slope fields. Here's an example. dy dx is equal to sine x and you see the graph of the slope field. Now the slope field itself is really the graph of the anti-derivative sine x, which is negative cosine x. But as you look at the slope field, there are many negative cosine x's in here. And if I pick out just one of these and follow the slopes along, I will get the graph of a particular solution to my derivative. So we not only have a slope field sitting here now, we have the graph of a particular solution. Well, how do we create these slope fields? That's what we're going to do in the next lesson. What we're going to do is dy dx is equal to x. That's the slope of some function and you probably already know what the function is. And we're going to get a piece of paper with grid lines on it or graph paper. We need to make a table first. So on our table, we only need the values of x because dy dx is equal to x. And so when we find dy dx, the only thing we have to do is say, well, if the value of x is zero, then the dy dx is equal to zero. If x is equal to one half, dy dx is equal to one half and so forth all the way down. And so we get this little table of values and we are going to graph their slopes. So we have when x was equal to zero, dy dx was equal to zero. So yes, it's equal to zero here, that slope. And if it's just a little line that you draw, it doesn't matter what the y value is, the slope is always zero. When x is equal to one half, the slope is always equal to one half. So we will draw all the one halves in. Again, it doesn't matter what y is. At one, the slopes are greater. So we draw them a little more sloped. And at negative one half, do the negative one half slope. And at negative one, a negative one slope. And at positive two, it's more sloped. And at negative two, it's more sloped. And as you look at the slope field, you see that it does look like a parabola. And if you remember from the last lesson, if dy dx is equal to x, then the function y is equal to x squared over two plus c, which is a parabola. And you'll see all the transformations of this parabola up and down the y-axis. We can also graph slope fields on our calculator. And in this session, I'm using the TI-83. What you need is the program called slope field. Put your function into y1, and then run the program. And it will graph the slope field as you see here, which is pretty similar to the one that we just had on our slide. So the next thing we want to learn how to do is to draw a particular function with an initial condition. So let's say we want the function where f of zero equals zero. This means we start at zero zero and follow the slope field along until we have a particular graph for the function that starts at zero zero. So this one would look something like this. Another initial condition could be f of zero equals negative one. And instead of starting at zero, this function is transformed down one, but it's the same function transformed. So you get your parabola that way. So we have just graphed a slope field with two different initial conditions. Let's try another slope field graph. This one we want to do is dy dx is equal to y. Now this looks very similar to the x one, but it's going to look a little bit different when we finish. Again, we have to make the table. So if we have y is equal to zero, dy dx will be equal to zero. And if y is equal to negative one half, dy dx is equal to negative one half, and so forth all the way down the table with the different values. Now this time it doesn't matter what x is, we are just looking at the y values. So on the grid paper, when y is equal to zero, dy dx is equal to zero. So this is all along this axis this way. When y is equal to one half, dy dx is equal to one half. So all the one halves on y's have a slope of one half. And when y is equal to one, the slope is one, and when y is equal to two, the slope is two. Now if dy dx were equal to negative one half, it looks like this, because all the y's are negative one half, and at negative one for both of them, and at negative two. And as you can see, this one is very different from the one we just finished, because it looks like there are two solutions here, are two types of solutions here, depending on whether you're above the x-axis or below the x-axis. And let's say I did f of zero equals one for a particular solution, that would be starting here and go up in that direction to the right, but when we come down, we will be heading towards the x-axis, it will be an asymptote. If we did f of zero equals negative one, then we would start at negative one, and again, do the right hand side, that one would go down, and then again towards the left, it would head towards the x-axis. Very different solution from the one you did before. You do not know how to do this in an anti-derivative form, but that is something you will be learning in the near future. Let's look at dy dx is equal to y on our calculator. So this time for our program, we put in a y, and it does take the y, don't worry about that. Again, run the program, and this time you will see it develop to something that we just did. And that is the solution for, or the slope field for dy dx is equal to y. Let's go on to our last example. This one's a little bit more complicated. Says on a piece of graph paper or grid, graph the derivative dy dx is equal to negative x over y. Now this time we need many more points. So what we're going to do is make a chart with quite a few points on it. And we need both x's and y's because we have x's and y's. So we need an x this time, a y, and then the dy dx. And this time dy dx is equal to negative x over y. And if we start out with x is equal to negative two, and do all those values of y, which are negative two, negative one, zero, one, and two, we will have, for the first one, negative two, negative two, dy dx is equal to negative one. And then for negative two, negative one, it is negative two. For negative two and zero, it isn't defined, but it's a vertical tangent. So let's put that in as a vertical tangent. For negative two and one, it's two. For negative two, two, it is one. So these are the slopes that we will graph. And let's just go to graphing this right away and then come back and do another set. If we graph these on our grid at negative two, negative two, we have a slope of negative one. At negative two, negative one, we have a slope of negative two. At negative two, zero, we have the vertical tangent line. At negative two, one, we have a slope of two. And at negative two, two, we have a slope of one. Well, let's go back and do some more. Let's say we do at two. Well, in looking all the values here, if we put a two in for x and a negative two in for y, the only thing we're doing is changing the sign on our slope. So we'll have a one, a two, another vertical tangent, a negative two and negative one. So let's put those in. So this would be a positive two, one vertical tangent. Go back to our chart and let's put a negative one in there. If we put negative two, negative one, zero, one and two again, a negative one, negative two would lead to a negative one half. A negative one, negative one would lead to a negative one. A negative one, zero again is the vertical tangent. A negative one, one would lead to one for our dy dx. And a negative one, two would be one half. So we would go from negative one half to one half. Let's put that one in. All right, at the negative one, negative two point, we would have negative one half. Negative two, negative one is negative one, the vertical tangent, the one, and then the negative one half. And then again, if we look at what happens at one, it's the same slope, only positive. So we can do that here, except for that vertical one. And we're beginning to see what this is and it should look like concentric circles. So if we actually did one with initial conditions where f of zero is equal to one, we should be able to draw a circle at one. If f of zero equals two, we have a circle with a radius of two. So we are able to draw all these concentric circles with our initial conditions. And again, we have the general solution of the slope fields, which is the whole slope field. And then we have the particular solution, f of zero equals one and f of zero is equal to two. Well, how would we do that on our calculator? Well, this time we have to put in for y equal to negative x divided by y and run the program. You will see those circles forming. So in this lesson, you have learned to draw slope fields and determine particular solutions. Many of the questions ask you to do this, but you'll also have matching questions which are just the general form on slope fields. This concludes your lesson on slope fields.