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## Differential equations

### Unit 1: Lesson 2

Slope fields- Slope fields introduction
- Worked example: equation from slope field
- Worked example: slope field from equation
- Worked example: forming a slope field
- Slope fields & equations
- Approximating solution curves in slope fields
- Worked example: range of solution curve from slope field
- Reasoning using slope fields

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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)

Given the slope field of a differential equation, we can sketch various solutions to the equation. In this example, we analyze the range of a specific solution.

## Video transcript

- [Instructor] If the initial
condition is zero comma six, what is the range of the solution
curve y is equal to f of x for x is greater than or equal to zero? 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 zero comma six, so zero comma six is
part of that solution. So let's see, zero comma six, so this is part of the solution. And we want to know the
range of the solution curve. So solution curve, you
can eyeball a little bit by looking at the slope field. So as x, remember, x is gonna be
greater than or equal to zero, so it's going to include
this point right over here. And as x increases, you
can tell from the slope, okay, y is gonna decrease, but it's gonna keep decreasing
at a slower and slower rate. And it looks like it's asymptoting towards the line y is equal to four. So it's gonna get really, as x gets larger and larger, larger, it's gonna get infinitely
close to y is equal to four, but it's not quite gonna get there. So the range, the y-values
that this is going to take on, y is going to be greater than four. It's never gonna be equal to four. So I'll do, it's going
to be greater than four. That's gonna be the
bottom end of my range. And at the top end of my
range, I will be equal to six. Six is the largest value
that I am going to take on. Another way I could have written
this is four is less than y is less than or equal to six. 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 'cause we're including x equaling zero. And then it keeps going
down, approaching four, getting very, very close to four, but never quite equaling four.