Slope Fields
Draw slope fields by evaluating dy/dx at a grid of points. Match slope fields to their differential equations. Sketch particular solutions through initial conditions. This is consistently tested on the AP exam.
A lesson of AP Calculus AB, in Differential Equations, about 10 minutes.
By the end of this lesson, you will be able to sketch and interpret slope fields for differential equations, match slope fields to their DEs, and use a slope field to sketch solution curves.
What Is a Slope Field?
A slope field (or direction field) is a graphical representation of a differential equation $\frac{dy}{dx} = f(x, y)$.
At each point $(x, y)$ in the plane, we draw a short line segment with slope equal to $f(x, y)$. The collection of these segments shows the "direction of flow" of solutions.
A slope field is a visual tool that helps us understand the family of solutions to a DE without actually solving it.
Constructing a Slope Field
For $\frac{dy}{dx} = f(x, y)$:
- Choose a grid of points $(x, y)$
- At each point, compute the slope: $m = f(x, y)$
- Draw a short segment with that slope centered at the point
Example: $\frac{dy}{dx} = x$
At $(0, 0)$: slope = 0 (horizontal segment)
At $(1, 0)$: slope = 1 (tilted up to the right)
At $(-1, 0)$: slope = -1 (tilted down to the right)
At any point $(x, y)$: slope = x (depends only on x, not y)
The slope field shows horizontal segments along the y-axis and steeper segments as $|x|$ increases.
Reading a Slope Field: Solution Curves
A solution curve (particular solution) is a curve that is tangent to the slope field at every point it passes through. It flows along the "grain" of the slope field.
Given an initial condition (e.g., the curve passes through $(1, 2)$), you can trace a solution curve by following the direction of the segments.
Example: $\frac{dy}{dx} = y$
| Point | Slope |
|---|---|
| $(0, 1)$ | 1 |
| $(0, 2)$ | 2 |
| $(0, -1)$ | -1 |
| $(1, 1)$ | 1 |
| $(1, 2)$ | 2 |
All segments at points with y > 0 slope upward; segments with y < 0 slope downward. The slope is steeper for larger |y|.
This DE has solution $y = Ce^x$ — exponential functions. If we trace any curve in the slope field starting above the x-axis, it curves upward exponentially.
Matching Slope Fields to DEs
Strategy: Look for patterns:
- Does the slope depend only on x? → slopes are the same across all y-values in a column
- Does the slope depend only on y? → slopes are the same across all x-values in a row