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# Approximating solution curves in slope fields

AP.CALC:
FUN‑7 (EU)
,
FUN‑7.C (LO)
,
FUN‑7.C.2 (EK)
,
FUN‑7.C.3 (EK)

## Video transcript

so we have the differential equation the derivative of Y with respect to X is equal to Y over six times four minus y and what we have plotted right over here is the slope field or a slope field for this differential equation and we can verify that this indeed is is a slope field for this differential equation let's draw a little table here so let's just verify a few points so say X Y and dy DX so let's say we start with I don't know let's start with this point right over here 1 comma 1 when X is 1 and Y is 1 well when I look at the differential equation 1 6 times 4 minus 1 so it's 1 6 times 3 which is 3 6 which is 1/2 and we see indeed on the slope field they depicted they depicted the slope there if a solution goes to that point and right at that point it's slope would be 1/2 and as you see it's actually only dependent on the Y value it doesn't matter what X is as long as Y is 1 dy DX is going to be 1/2 and you see that's why when X is 1 and 1/2 and Y is 1 you still have a slope of 1/2 and 4 as long as Y is 1 all of these all of these sampled points right over here all have a slope of 1/2 so that that's you know we're just looking at that that makes us feel that this that this slope field is consistent with this differential equation but let's try a few other points just to feel a little bit better about it and then we will use the slope field to actually visualize some solutions so let's say added let's do an interesting point let's say we have this point actually no that's that's it that's it a half point let's say we have this this let's see I want to do let's say we do this point right over here so that's X is equal to 1 and Y is equal to 6 and we see the way the differential equation is defined it doesn't matter what our X is it's really dependent on the Y that's going to drive the slope but we have 6 over 6 which is 1 times 4 minus 6 which is negative 2 so it's negative 2 so we should have a slope of negative 2 and it looks like that's what they depicted so as long as Y is 6 we should have a slope of negative to have a slope of negative two and you see that in the slope field so hopefully you feel pretty good that this is the slope field for this differential equation if you don't I encourage you to keep keep verifying these points here but now let's actually use the slope field let's actually use this to visualize solutions to this differential equation based on points that the solution might go through so let's say that we have a solution that goes through this point right over here so what is that solution likely to look like and if once again it's going to be a rough approximation well right at that point it's going to have a slope just as a slope field shows and as our Y increases it looks like our slope it looks like our slope so at this point I should be actually let me let me undo that so this if I keep going up at this point when Y is equal to two I should be parallel to all of these these segments on the slope field at Y is equal to two and then it looks like the slope starts to decrease as as we approach Y is equal to four and so if I had a solution that went through this point my guess is that it would look something and then now the slope decreases again as we as well as we approach Y is equal to zero and of course we see that because if when y equals zero this whole thing is zero so our derivative is going to be zero so a reasonable solution might look something like this so this gives us a clue well look if a solution goes through this point this right over here might might be what it looks like but what if it goes through I don't know what if it goes through this point right over here well then it might look like it might look like this by the same exact logic so it might look like this so just like that we're starting to get a sense we don't know the actual solution for this differential equation but we're trying to get a sense of what what type of functions what type of functions are the class of functions that might satisfy the differential equation but what's interesting about this slope field it looks like there's some you know there's some interesting stuff you know any if we have if our solution includes points between where the y values between 0 and 4 it looks like we're gonna have solutions like this but what if we had Y values that were larger than that or that were less than that or exactly exactly 0 or 4 so for example what if we had a solution that went through this point right over here well at that point right over here the slope field tells us that our slope is zero so our Y value is not going to change it as long as our Y value doesn't change our Wi-Fi was going to stay at 4 so our slope is going to stay 0 so we actually already found this is actually a solution to the differential equation it's y is equal to 4 is a solution to this differential equation so Y is equal to 4 y is equal to 4 and you can verify that that is a solution when y is equal to 4 this right hand side is going to be 0 and the derivative is 0 for y is equal to 4 so that is a solution to the differential equation and the same thing for y is equal to 0 that is also a solution to the differential equation now what if we included points what if we included this point up here and actually let me do it in a different color so that you could see it let's say our solution include at that point well then it might look something it might look something like this and once again I'm just using the slope field as a guide to give me an idea of what the slope might be as my curve progresses as my solution progresses so a solution that includes the point 0 5 might look something like this and once again it's just another clue a solution that includes the point 0 negative 1 and 1/2 might look something like might look something like this so anyway hopefully this gives you a better appreciation for why slope fields are interesting if you have a differential equation that just involves the first derivative and some X's and Y's this one only involves the first derivative and Y's we can plot a slope field like this is not too much trouble if we since you just keep solving for the slopes and then we can use that slope field to get a conceptual or visual understanding of what the solutions might look like given points that the solutions might actually contain