Key Idea: A slope field draws dy/dx = f(x, y) as a grid of short tangent segments. It lets you see and sketch solution curves without solving the equation.
| Task | How |
|---|---|
| Gradient at a point | Substitute (x, y) into f |
| Isocline f = c | All segments there are parallel, with gradient c |
| Flat segments | Solve f(x, y) = 0: max or min of curves |
| Sketch a solution | Start at the point and stay tangent |
Reading a solution curve
- Increasing where f > 0.
- Decreasing where f < 0.
- Turning where f = 0.
A solution curve simply follows the little segments.
dy/dx = x − y. Gradient at (3, 1)? 2.
Where are all the segments flat? On the line y = x.
A curve starts at (0, 2). Does it rise or fall at first? Fall: f(0, 2) = −2.
What happens where that curve meets y = x? Its gradient is 0, so it has a minimum there, then rises.