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Slope fields introduction
- [Voiceover] Let's say that we have the differential equation dy dx or the derivative of y with respect to x is equal to negative x over y. Let's say we don't know how to find the solutions to this, but we at least want to get a sense of what the solutions might look like. And to do that what we could do is we could look at a coordinate plane, so let me draw some axes here. So let me draw relatively straight line alright. So that's my y-axis and this is my x-axis. Let me mark this as one, that's two, that's negative one, negative two, one, two, negative one, and negative two. And what I could do is since this differential equation is just in terms of xs and ys and first derivatives of y with respect to x, I could sample points on the coordinate plane, I could look at the x and y coordinates substitute them in here, figure out what the slope is going to be. And then I could visualize the slope if a solution goes through that point what the slope needs to be there and I can visualize that with a line segment, a little small line segment that has the same slope as the slope in question. So let's actually do that. So let me setup a little table here. I'm going to do a little table here to do a bunch of x and y values. Once again I'm just sampling some points on the coordinate plane to be able to visualize. So x, y and this is dy dx. So let's say when x is zero and y is one, what is the derivative of y with respect to x? It's going to be negative zero over one so it's just going to be zero. And so at the point zero one if a solution goes through this point its slope is going to be zero. And so we can visualize that by doing a little horizontal line segment right there. So let's keep going. What about when x is one and y is one? Well then dy dx, the derivative of y with respect to x is negative one over one, so it's going to be negative one. So at the point one comma one if a solution goes through that point, it would have a slope of negative one. And so I draw a little line segment that has a slope of negative one. Let me do this in a new color. What about when x is one and y is zero? Well then it's negative one over zero, so this is actually undefined, but it's a clue that maybe the slope there if you had a tangent line at that point, maybe it's vertical. So I'll put that as a question mark, vertical there. So maybe it's something like that if you actually did have, I guess it wouldn't be a function, if you had some kind of relation that went through it, but let's not draw that just yet. But let's try some other points. Let's try the point negative one negative one. So now we have negative negative one, which is one, over negative one. Well you'd have a slope of negative one here. So negative one negative one you'd have a slope of negative one. What about if you had one negative one? Well now it's negative one over negative one, your slope is now one. So one negative one, if a solution goes through this its slope would look like that. And we could keep going, we could even do two negative two. That's going to have a slope of one as well. If you did positive two positive two, that would be negative two over two. You'd have a slope of negative one right over here. And so we could do a bunch of points, just keep going. Now I'm just doing them in my head, I'm not going on the table. But you get a sense of what's going on here. Here you're slope, what if it was negative one one? It's going to have a slope of one. So at this point your slope negative one one so negative negative one is one over one, so you're going to have a slope like that. At negative two two same exact idea, it would look like that. And so when you keep drawing these line segments over these kind of sample points in the cartesian or in the x-y plane, you start to get a sense of well what would a solution have to do. And you can start to visualize that hey maybe a solution would have to do something like this. This would be a solution, so maybe it would have to do something like this. Or if we're looking only at functions and not relations I'll make it so it's very clear. So maybe it would have to do something like this. Or if the function started here based on what we've seen so far maybe it would have to do something like this. Or if this were a point on the function over here, it would have to do something like this. And once again I'm doing this based on what the slope field is telling me. So this field that I'm creating where I'm sampling a bunch of points and I'm visualizing the slope with a line segement. Once again this is called a slope field. So hopefully that gives you kind of the basic idea of what a slope field is. And the next two videos we'll explore this idea even deeper.