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# Worked example: range of solution curve from slope field

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

## Video transcript

if the initial condition is 0 comma 6 what is the range of the solution curve y is equal to f of X for X is greater than or equal to 0 so we have a slope field here for a differential equation and we're saying okay if we have a solution where the initial condition is 0 comma 6 so 0 comma 6 is part of that solution so let's see 0 comma 6 so this is part of the solution and we want to know the range of the solution curve so a solution curve you can eyeball a little bit by looking at the slope field so as X remember X is going to be greater than or equal to 0 so it's going to include this point right over here and as x increases you can tell from the slope ok y is going to decrease but it's going to keep decreasing at a slower and slower rate and it looks like it's asymptoting towards the line y is equal to 4 so it's going to get really it's as X gets larger and larger larger it's going to get infinitely close to Y is equal to 4 but it's not quite going to get there so the range the Y values that this is going to take on y is going to be greater than 4 it's not ever going to be equal to 4 so I'll do it's going to be greater than 4 that's going to be the bottom end of my range and at the top end of my range I will be equal to 6 6 is the largest value that I am going to take on another way I could have written this is 4 is less than Y is less than or equal to 6 either way this is a way of describing the range the Y values that the solution will take on for X being greater than or equal to 0 if they said for all X's well then you might have been able to go back this way and keep going but they're saying the range of the solution curve for X is greater than or equal to 0 so we won't consider those those values of x less than 0 so there you go the curve would look something like that and you can see the highest value it takes on is 6 and it actually does take on that value because we're including X equaling a 0 and then it keeps going down approaching 4 getting very very close to 4 but never quite equaling 4