If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Worked example: slope field from equation

AP.CALC:
FUN‑7 (EU)
,
FUN‑7.C (LO)
,
FUN‑7.C.1 (EK)

## Video transcript

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&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 they'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 going to have I'm going to 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 2 and Y is 2 when X is 2 and Y is 2 the derivative of Y with respect to X is going to be 2 minus 2 it's going to be equal to 0 and just with that let's see here this this slope on this slope field does not look like it's 0 this looks like it's negative 1 so already I could rule this one out this slope right over here looks like it's positive 1 so I'll rule that out it's definitely not 0 this slope also looks like positive 1 so I can rule that one out this slope at 2 comma 2 actually does look like a zero so I'm liking this one right over here this slope at 2 comma 2 looks larger than 1 so I could rule that out so it was that straightforward to deduce that this 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 1 whenever X is equal to Y you're going to get the derivative equaling 0 and you see that here when you at44 derivative equals 0 when it's six-six derivative equals 0 at negative 2 negative 2 derivative equals 0 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 4 y is 2 then the derivative here should be four minus two which is going to be two so when X is 4 y is 2 we do indeed see that the slope field is indicating a slope that looks like 2 right over here and if it was the other way around when X is when X is let's say X is negative 4 and Y is negative 2 so negative 4 negative 2 well negative 4 minus negative 2 is going to be negative 2 and you can see that right over here negative 4 negative 2 you can see the slope right over here it's a little harder to see looks like negative 2 so once again using even just this first 2 comma 2 coordinates we were able to deduce that this was the choice but it just continues to confirm our original answer
AP® is a registered trademark of the College Board, which has not reviewed this resource.