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# Slope fields introduction

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

## Video transcript

let's say that we have the differential equation dy/dx of 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 all right so that's my y-axis and this is my x-axis let me do draw let me mark this as 1 that's 2 that's negative 1 negative 2 1 2 negative 1 and negative 2 and what I could do is since this differential equation is just in terms of X's and Y's and first derivatives of Y with respect to X I could go 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 I could visualize the slope if a solution goes to 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 set up a little table here let me 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 let's say when x is 0 and Y is 1 what is the derivative of Y with respect to X it's going to be negative 0 over 1 so it's just going to be 0 and so at the point 0 1 if a solution goes through this point it's slope is going to be 0 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 1 and Y is 1 well then dy DX the derivative of Y with respect to X is negative 1 over 1 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 it would have a slope of negative one and so I draw a little line segment that has a slope of negative one what about when X is 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 maybe the slope there I guess we're if you had a tangent line at that point maybe it's vertical so so I'll put that as a question mark vertical there and 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 just put let's not draw that just yet but let's try some other points let's say that we had let's try the point negative one negative one so now we have negative negative one which is 1 over negative 1 well you would have a slope of negative 1 here so negative 1 negative 1 you would have a slope of you would have a slope of negative 1 what about if you had one negative 1 well now it's negative 1 over negative 1 your slope is now 1 so 1 negative 1 if your solution if a solution goes through this its slope would look like that and we could keep going we could even do 2 negative 2 that's going to have a slope of 1 as well if you did if you did positive 2 positive 2 that be negative 2 over 2 you'd have a slope of negative 1 right over here and so we could do a bunch of we could do a bunch of points just keep going up now I'm not 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 your slope would if is negative 1 1 it's going to have a slope of 1 so at this point your slope negative 1 1 so negative negative 1 is 1 over 1 so you have slope like that at negative 2 - same exact idea it would like that and so you get a when you keep drawing these line segments over at these kind of these sampled points in the Cartesian or in the in the XY 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 a solution would have to do something something like this this would be a a solution so maybe it would have to do something like this or if we're looking if we're looking only at functions and not relations you know I'll only I'll make it so it's a very clear so maybe we would have to do something like this or if a if the function started like here based on what we've seen so far maybe we would have to do something maybe we would have to do something like this or if it was 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 taking I'm sampling a bunch of points and I'm visualizing the slope with a line segment once again this is called a slope slope field so hopefully that gives you kind of the basic idea of what a slope field is in the next few videos we'll explore this idea even deeper
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