01 / 21

Sketching
Slope Fields

Learn to visualize differential equations by drawing slope fields: a grid of tiny line segments whose slopes are given by \(\frac{dy}{dx}\).

AP Calculus BC

What Is a Slope Field?

A slope field (also called a direction field) is a graphical representation of a first-order differential equation \(\frac{dy}{dx} = f(x, y)\). At each sample point \((x, y)\) in the plane, you draw a short line segment whose slope equals \(f(x, y)\).

Why Do Slope Fields Matter?

★ Visual Insight
Many DEs have no closed-form solution. Slope fields let you see the behavior of solutions qualitatively.
★ Initial Conditions
Given an initial value \(y(x_0) = y_0\), you can trace a curve through the field to approximate the particular solution.
★ AP Exam Skill
The AP Calculus BC exam frequently asks you to sketch or match slope fields and identify solution curves.

Anatomy of a Slope Field

x y Each segment has slope = \(f(x,y)\) slope here

The Core Rule

1
Evaluate \(f(x, y)\) at the point to get the slope \(m\).
2
Draw a tiny line segment centered at \((x, y)\) with that slope.
3
Repeat for all grid points to see the global pattern.

Special Slopes to Look For

Horizontal Segments
When \(f(x,y) = 0\), the segment is flat. The set of points where this happens forms the nullcline.
Vertical Tendency
When \(|f(x,y)| \to \infty\), segments become nearly vertical: the solution curve is changing very rapidly.
Constant Slopes
Along curves where \(f(x,y) = c\) (isoclines), every segment has the same slope \(c\).

Slope Field vs Solution Curve

Slope Field Solution Curve
What it is A grid of short segments A continuous function \(y(x)\)
Information Shows slope at many points Shows one particular solution
Requires Only the DE: \(\frac{dy}{dx}=f(x,y)\) DE + initial condition \(y(x_0)=y_0\)
Uniqueness One field per DE Infinitely many curves (one per IC)
Relationship Each solution curve is tangent to every segment it passes through

Using Isoclines to Speed Up Sketching

An isocline is a curve along which \(f(x, y) = c\) for some constant \(c\). Along an isocline, every slope segment has the same tilt. This makes sketching much faster: find the isocline, then draw parallel segments along it.

Step-by-Step: Sketching a Slope Field

1
Set up a grid. Choose a window (e.g., \(-3 \le x \le 3\), \(-3 \le y \le 3\)) and mark integer grid points.
2
Evaluate slopes. At each point \((x_i, y_j)\), compute \(m = f(x_i, y_j)\).
3
Identify isoclines. Find where \(f(x,y)=0\), \(f(x,y)=1\), \(f(x,y)=-1\), etc. Draw all segments along each isocline with the same slope.
4
Draw segments. At each grid point, draw a short centered line with slope \(m\). Keep all segments the same length.
5
Check patterns. Look for horizontal segments (nullcline), symmetry, and regions where slopes increase or decrease.

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

x y 1 -1 1 -1 m=0 m=1 m=-1

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

x y y = x

Example 3: \(\frac{dy}{dx} = x^2 - 1\)

x y x = -1 x = 1

Tracing a Solution Curve

Once a slope field is drawn, you can sketch a particular solution by starting at an initial point \((x_0, y_0)\) and following the segments: the solution curve is always tangent to the local segment.

1
Place your pencil at \((x_0, y_0)\).
2
Follow the direction of the local segment, drawing a smooth curve.
3
Never cross a segment at a sharp angle: the curve should flow smoothly with the field.

Recognizing Patterns in Slope Fields

★ \(f\) depends only on \(x\)
Segments in the same vertical column are parallel.
★ \(f\) depends only on \(y\)
Segments in the same horizontal row are parallel.
★ \(f\) depends on \(x\) and \(y\)
Slopes vary in both directions: look for diagonal nullclines and changing steepness.

Common Mistakes

✕ Varying Segment Lengths
All segments should be the same length. Longer segments for steeper slopes misleads: length encodes no information; only angle matters.
✕ Connecting Segments
Segments are not part of a curve. They float at each grid point. Don't draw them end-to-end like a piecewise line.
✕ Forgetting Negative Slopes
When \(f(x,y) < 0\), the segment tilts downward left-to-right. Accidentally drawing all segments upward is a common error.

Summary: Slope Field Checklist

Feature What to Do Watch For
Grid Points Use integer coordinates Don't skip the origin
Nullcline Set \(f(x,y)=0\), sketch the curve All segments are horizontal here
Isoclines Set \(f(x,y)=c\) for key values Parallel segments along each
Sign of \(f\) Check above/below nullcline Positive = upward tilt
Symmetry Check if \(f\) is even, odd, etc. Reduces work by half
Solution Curves Trace smoothly, always tangent Curves never cross

Question 1 of 5

If \(\frac{dy}{dx} = y - 2\), at which value of \(y\) are the slope segments horizontal?

Question 2 of 5

A slope field has all segments in each vertical column identical. Which type of differential equation could produce this field?

Question 3 of 5

For the slope field of \(\frac{dy}{dx} = x^2 + y^2\), what can you conclude about all slope segments?

Question 4 of 5

When sketching a solution curve through a slope field starting at \((0, 1)\), which statement is true?

Question 5 of 5

The nullcline of \(\frac{dy}{dx} = 2x - y + 3\) is the line: