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Worked example: range of solution curve from slope field | AP Calculus AB | Khan Academy

2m read
·Nov 11, 2024

If the initial condition is (0, 6), what is the range of the solution curve ( Y = F(x) ) for ( x \geq 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, 6), so (0, 6) is part of that solution.

Let's see (0, 6). So this is part of the solution, and we want to know the range of the 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 zero, so it's going to include this point right over here. As ( x ) increases, you can tell from the slope, okay, ( y ) is going to decrease, but it's going to keep decreasing at a slower and slower rate.

It looks like it's asymptoting towards the line ( y = 4 ). So, it's going to get really, as ( x ) gets larger and larger, it's going to get infinitely close to ( y = 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.

Six is the largest value that I am going to take on. Another way I could have written this is ( 4 < y \leq 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 zero.

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 zero.

So we won't consider those values of ( x ) less than zero. So there you go, the curve would look something like that, and you can see the highest value it takes on is six, and it actually does take on that value because we're including ( x ) equaling zero, and then it keeps going down, approaching 4, getting very, very close to 4 but never quite equaling 4.

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