Reasoning Using Slope Fields

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.

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.

Each segment of a slope field is the local slope; a solution curve follows them

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.

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.

Write from a curve that cannot cross

• A slope field has an equilibrium where the dashes are horizontal.
• Two solution sketches are started at different heights.