Reasoning Using Slope Fields · 利用方向场推理
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| solution curve/səˈluːʃn kɜːv/ | 解曲线 | jiě qū xiàn |
| long-term behavior/lɒŋ tɜːm bɪˈheɪvjə/ | 长期行为 | cháng qī xíng wéi |
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.
跟着箭头画出曲线
- 斜率场是一组切线方向;解曲线在其中穿行。
- 从一个给定点出发,移动时始终与附近的短线保持相切。
- 你描出的曲线就是过该点的特解。
- 无需公式——你直接从场中推理出解。
A solution curve through the slope field must always stay... · 穿过方向场的解曲线必须始终...
It follows the tangent directions. · 它遵循切线方向。
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.
过一点画出解曲线
- 把笔放在给定点,跟随局部斜率,随短线转向而弯曲。
- 绝不以错误角度穿过短线——始终沿着它们走。
- 不同的起点给出解族的不同成员。
- 短线像水流,你的曲线是随之漂流的小船。
Thread a curve through the field · 在方向场中穿引曲线
A solution curve stays tangent to every dash it passes — drag the start point and watch it follow the field. · 解曲线经过的每一段短线都与其相切——拖动起点观察其如何跟随场。
Long-term behavior from the pattern
- Look far right (large $x$ or $t$) and read the long-term behavior 长期行为 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.
从图案看长期行为
- 看远处右侧(大的 $x$ 或 $t$),读出解的长期行为。
- 短线是否朝一条水平线变平?解趋近那个值(一个平衡)。
- 它们是否永远向上变陡?解无界增长。
- 场的整体形状预测解最终去向哪里。
The overall pattern of a slope field forecasts the ____ behavior of solutions. · 方向场的整体图案预测了解曲线的____行为。
Read where the dashes send solutions as $x$ grows. · 读取随着$x$增长，短线将解导向何处。
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.
平衡:稳定或不稳定
- 一行水平短线是一个平衡——一个常数解。
- 若附近的解朝它弯去,该平衡是稳定的(吸引子)。
- 若它们弯离,则是不稳定的。
- 读箭头指向那条线还是背离它,就知道是哪种。
Two different solution curves of the same equation can cross each other. · 同一方程的两条不同解曲线可能相交。
Each point has one slope, so solution curves never cross. · 每一点只有一个斜率，因此解曲线永不相交。
For · 支持 $\dfrac{dy}{dt}=y$, the equilibrium $y=0$ is... · 对于$\dfrac{dy}{dt}=y$，平衡点$y=0$是...
Above $0$ slopes push up, below they push down → away from $0$. · 在$0$上方斜率向上推，下方斜率向下推 → 远离$0$。
A horizontal row of dashes in a slope field represents a... · 方向场中的一行水平短线代表一个...
Slope · 斜率 $0$ everywhere along it → a constant solution. · 沿此线处处斜率为$0$ → 常数解。
Select all · 所有 true statements about equilibria. · 选择关于平衡点的所有正确陈述。
Equilibria can be at any $y$ where the rate is $0$; toward = stable, away = unstable. · 平衡可以出现在任何$y$处，其中速率是$0$；朝向=稳定，远离=不稳定。
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.
解曲线必须与斜率场保持相切——它跟随短线,而不横切它们。而且同一方程的两条不同解曲线永不相交(每点只有一个斜率)。若你的草图有曲线相交,你就画错了。
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.
对 $\dfrac{dy}{dt}=y$,描述过 $(0,1)$ 与 $(0,-1)$ 的解。
- 直线 $y=0$ 是一个平衡(那里斜率 $0$)。
- 在它上方($y>0$),斜率为正 → 过 $(0,1)$ 的解增长,远离 $0$。
- 在它下方($y<0$),斜率为负 → 过 $(0,-1)$ 的解下落,远离 $0$。
- 所以 $y=0$ 是一个不稳定平衡——解都逃离它。
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.
要用斜率场推理,过一个给定点、始终与短线相切地画出一条解曲线。场的整体图案揭示长期行为——解可能趋近一个平衡(稳定)或逃离(不稳定)。解曲线永不相交。