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# Worked example: forming a slope field

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

## Video transcript

in drawing the slope field for the differential equation the derivative of Y with respect to X is equal to y -2 X I would place short line segments at select points on the XY plane complete the sentences at the point negative 1 comma 1 I would draw a short segment of slope blank and like always pause this video and see if you can fill out these three blanks well when your the short segments that you're trying to draw to construct this slope field you figure out their slope based on the differential equation so you're saying when X is equal to negative 1 and Y is equal to 1 what is the derivative of Y with respect to X and that's what this differential equation tells us so for this first case the derivative of Y with respect to X is going to be equal to Y which is 1 minus 2 times X X is negative 1 so this is going to be negative 2 but you're subtracting it so it's going to be plus 2 so the derivative of Y with respect to X at this point is going to be 3 so I would draw a short line segment or a short segment of slope 3 and we keep going at the point 0 comma 2 well let's see when x is 0 and Y is 2 the derivative of Y with respect to X is going to be equal to Y which is 2 minus 2 times 0 well that's just going to be 2 and then last but not least for this third point the derivative of Y with respect to X is going to be equal to Y which is 3 minus 2 times X X here is 2 2 times 2 3 minus 4 is equal to 4 3 minus 3 minus 4 is equal to negative 1 and that's all that problem asks us to do now if we actually had to do it it would look something like I'll try to draw it real fast so let's see make sure I can go to let make sure have space for all of these points here so so that's my my coordinate axes and I want to get the point 0 comma 2 so that's 0 comma 2 actually I want to go all the way to 2 comma 3 so let me get some space here so 1 2 3 and then 1 2 3 and then we have to go negative 1 comma 1 so you might go we might go right over here and so for this first one and this exercise isn't asking us to do it but I'm just making it very clear how we would construct the slope field so the point negative 1 comma 1 negative 1 comma 1 a sort segment of slope 3 so slope 3 would look something like something like that then at the point at the point 0 comma 2 a slope of 2 0 comma 2 the slope is going to be 2 which looks something like that and then at the point 2 comma 3 at 2 comma 3 a short segment of slope negative 1 so 2 comma 3 a slope a segment of slope negative 1 it would look something like that and you would keep doing this at more and more points if you had a computer to do it that's what the computer would do and you would draw these sorts line segments to indicate what the derivative is at those points and you get a sense of the solution space for that differential equation
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