- Slope fields introduction
- Worked example: equation from slope field
- Worked example: slope field from equation
- Worked example: forming a slope field
- Slope fields & equations
- Approximating solution curves in slope fields
- Worked example: range of solution curve from slope field
- Reasoning using slope fields
Given a differential equation in x and y, we can draw a segment with dy/dx as slope at any point (x,y). That's the slope field of the equation. See how we match an equation to its slope field by considering the various slopes in the diagram.
- [Instructor] Which slope field is generated by the differential equation the derivative of y with respect to x is equal to x minus y? And like always pause this video and see if you can figure it out on your own. Well the easiest way to think about a slope field, if I was, if i needed to plot this slope field by hand, I would sample a bunch of x and y points. And then I would figure out what the derivative would have to be at that point. And so what we can do here, since we've already drawn some candidate slope fields for us, is figure out what we think the slope field should be at some points and see which of these diagrams, these graphs, or these slope fields actually show that. So let's, let me make a little table here, so I'm gonna have, x, y and then the derivative of y with respect to x. And we can do it at a bunch of values, so let's think about it. Let's think about when, we're at this point right over here. When x is two and y is two. When x is two and y is two, the derivative of y with respect to x is going to be two minus two. It's going to be equal to zero. And just with that, let's see, here this slope on this slope field does not look like it's zero. This looks like it's negative one. So already I can rule this one out. This slope right over looks like it's positive one. So I'll rule that out, it's definitely not zero. This slope also looks like positive one. So I can rule that one out. This slope at two comma two actually does look like zero. So I'm liking this one right over here. This slope at two comma two looks larger than one so I can rule that out. So it was that straight forward to deduce that this choice right over here is, if any of these are going to be the accurate slope field, it's this one. But just for kicks we could keep going to verify that this is indeed the slope field. So let's think about what happens when x is equal to a one, whenever x is equal to y, you're gonna get the derivative equaling zero. And you see that here, when you're at four, four, derivative equals zero. When it's six, six, derivative equals zero. At negative two, negative two, derivative equals zero. So that feels good, that this is the right slope field. And then we could pick other arbitrary points. Let's say when x is four, y is two, then the derivative here should be four minus two, which is going to be two. So when x is four, y is two, we do indeed see that the slope field is indicating a slope that looks like two right over here. And if it was the other way around, when x is let's say, x is negative four and y is negative two. The negative four, negative two. Well negative four minus negative two is going to be negative two. And you can see that right over here. Negative four, negative two, you can see the slope right over here. It's a little harder to see, looks like negative two. So once again using even just this first two comma two coordinates we were able to deduce this was the choice. But it just continues to confirm our original answer.