Reasoning Using Slope Fields
Introduced| English |
|---|
| solution curve/səˈluːʃn kɜːv/ |
| long-term behavior/lɒŋ tɜːm bɪˈheɪvjə/ |
Following the arrows to a curve
- A slope field is a set of tangent directions; a solution curve 解曲线 threads through them.
- Start at a given point and move so you always stay tangent to the nearby dashes.
- The curve you trace is the particular solution passing through that point.
- No formula needed — you reason about solutions straight from the field.
A solution curve through the slope field must always stay...
It follows the tangent directions.
Sketching a solution through a point
- Put your pencil at the given point and follow the local slope, curving as the dashes turn.
- Never cross the dashes at the wrong angle — always run along them.
- Different starting points give different members of the solution family.
- The dashes act like a current, and your curve is a boat drifting with it.
Thread a curve through the field
A solution curve stays tangent to every dash it passes — drag the start point and watch it follow the field.
Long-term behavior from the pattern
- Look far right (large $x$ or $t$) and read the long-term behavior 长期性态 of solutions.
- Do the dashes flatten toward a horizontal line? Solutions approach that value (an equilibrium).
- Do they steepen upward forever? Solutions grow without bound.
- The overall shape of the field forecasts where solutions end up.
The overall pattern of a slope field forecasts the ____ behavior of solutions.
Read where the dashes send solutions as $x$ grows.
Equilibria: stable or unstable
- A horizontal row of dashes is an equilibrium — a constant solution.
- If nearby solutions curve toward it, the equilibrium is stable (an attractor).
- If they curve away, it's unstable.
- Reading whether arrows point toward or away from the line tells you which.
Two different solution curves of the same equation can cross each other.
Each point has one slope, so solution curves never cross.
For $\dfrac{dy}{dt}=y$, the equilibrium $y=0$ is...
Above $0$ slopes push up, below they push down → away from $0$.
A horizontal row of dashes in a slope field represents a...
Slope $0$ everywhere along it → a constant solution.
Select all true statements about equilibria.
Equilibria can be at any $y$ where the rate is $0$; toward = stable, away = unstable.
A solution curve must stay tangent to the slope field — it follows the dashes, it doesn't cut across them. And two different solution curves for the same equation never cross (each point has just one slope). If your sketch has curves intersecting, you've drawn it wrong.
For $\dfrac{dy}{dt}=y$, describe solutions through $(0,1)$ and $(0,-1)$.
- The line $y=0$ is an equilibrium (slope $0$ there).
- Above it ($y>0$), slopes are positive → the solution through $(0,1)$ grows away from $0$.
- Below it ($y<0$), slopes are negative → the solution through $(0,-1)$ falls away from $0$.
- So $y=0$ is an unstable equilibrium — solutions run away from it.
To reason with a slope field, sketch a solution curve by staying tangent to the dashes through a given point. The field's overall pattern reveals long-term behavior — solutions may approach an equilibrium (stable) or run away (unstable). Solution curves never cross.