01 / 21

Reasoning Using
Slope Fields

Visualize differential equations by sketching direction fields and tracing solution curves through families of tangent segments.

AP Calculus AB

What Is a Slope Field?

A slope field (also called a direction field) is a grid of short line segments drawn at sample points \((x, y)\) in the plane. Each segment has slope equal to \(\frac{dy}{dx}\) evaluated at that point.

Together, these tiny segments show the direction a solution curve would travel if it passed through each point: no algebra required.

Anatomy of a Slope Field

Why Slope Fields Matter

Visualize the Unsolvable
Many DEs have no closed-form solution. Slope fields let us see the behavior of every solution at once.
Identify Equilibria
Horizontal segments reveal where \(\frac{dy}{dx}=0\): these are equilibrium solutions (constant functions).
Predict Long-Term Behavior
Without solving, you can tell whether solutions grow, decay, oscillate, or approach an asymptote.

The Core Idea: Tangent Segments

Given a differential equation \(\frac{dy}{dx} = f(x,y)\):

1
Pick a grid point \((x_0, y_0)\).
2
Compute the slope \(m = f(x_0, y_0)\).
3
Draw a short segment through \((x_0, y_0)\) with slope \(m\).
4
Repeat for every point in the grid.

Reading a Slope Field

Slope Depends on \(x\), \(y\), or Both

\(\frac{dy}{dx}=f(x)\)
Slopes are the same along every vertical line (columns look identical).
\(\frac{dy}{dx}=g(y)\)
Slopes are the same along every horizontal line (rows look identical). These are autonomous DEs.
\(\frac{dy}{dx}=h(x,y)\)
Slopes vary in both directions: no obvious row or column pattern.

Key Concept: Isoclines

An isocline is a curve along which the slope is constant. Setting \(f(x,y) = c\) for a constant \(c\) gives the isocline of slope \(c\).

1
The nullcline is the isocline where \(c = 0\): segments are horizontal.
2
Along a \(c = 1\) isocline, every segment has slope \(1\) (tilted at \(45^\circ\)).
3
Drawing several isoclines quickly reveals the global structure of the field.

Key Features to Analyze

When matching a slope field to a differential equation, look for these key structural features first. They quickly eliminate incorrect choices.

Nullclines
Where are segments horizontal? This happens when \(\frac{dy}{dx} = 0\). Find the curve(s) where this occurs.
Row/Column Patterns
Are segments in each row parallel? Then \(\frac{dy}{dx}\) depends only on \(y\). In each column? Only on \(x\).
Sign Analysis
In which quadrants or regions are slopes positive (rising) vs. negative (falling)? Check a test point in each region.

How to Sketch a Slope Field

1
Set up a grid of sample points (e.g., integer coordinates from \(-3\) to \(3\)).
2
Evaluate \(f(x,y)\) at each point to find the slope.
3
Find the nullcline (\(\frac{dy}{dx}=0\)): draw horizontal dashes there first.
4
Classify regions: where is slope positive? Negative? Large? Small?
5
Draw segments with appropriate angle. Keep them short and uniform in length.
6
Trace a solution curve from an initial condition by following the flow.

Example 1: \(\frac{dy}{dx} = y\)

Example 2: \(\frac{dy}{dx} = -\frac{x}{y}\)

Example 3: \(\frac{dy}{dx} = x - y\)

Matching a DE to Its Field

1
Check dependence: Are column segments identical (\(f(x)\) only)? Row segments identical (\(g(y)\) only)?
2
Find the nullcline: Where are segments horizontal? Set \(f(x,y) = 0\) and check.
3
Test a specific point: Plug in a known \((x,y)\) and verify the segment slope matches \(f(x,y)\).
4
Check signs: Where should slopes be positive/negative? Compare to the field.

Common Mistakes

Varying Segment Lengths
All segments should be the same length. The slope sets the angle, not the length.
Crossing Solution Curves
Solutions to a first-order DE never cross (uniqueness theorem). If your curves intersect, something is wrong.
Ignoring Undefined Points
If \(f(x,y)\) is undefined at a point (e.g., division by zero), do not draw a segment there.

Summary: Slope Field Checklist

Feature What to Look For What It Tells You
Horizontal segments Nullcline: \(f(x,y) = 0\) Equilibria or turning points
Vertical segments \(f(x,y) \to \pm\infty\) Slope is undefined or very steep
Parallel rows DE depends only on \(y\) Autonomous equation
Parallel columns DE depends only on \(x\) Separable and easy to integrate
Converging segments Slopes approach each other Stable equilibrium nearby
Diverging segments Slopes fan apart Unstable equilibrium nearby

Question 1 of 5

In a slope field for \(\frac{dy}{dx} = y - 2\), at which value of \(y\) are all segments horizontal?

Question 2 of 5

If a slope field has identical segments in every row (same slope along horizontal lines), the differential equation depends on:

Question 3 of 5

For the DE \(\frac{dy}{dx} = x + y\), what is the slope of the segment at the point \((1, -3)\)?

Question 4 of 5

Two solution curves in a slope field for a first-order DE can never cross each other. This is a consequence of:

Question 5 of 5

In the slope field for \(\frac{dy}{dx} = -\frac{x}{y}\), the solution curves are: