AP Calculus AB · Unit 7: Differential Equations

7.4 Reasoning Using Slope Fields

Sketching a solution

To sketch a particular solution through a given point, start there and follow the segment directions. The curve stays tangent to the local segments.

Reading behavior

Horizontal segments mean dy/dx = 0. An equilibrium (constant) solution occurs where dy/dx = 0 for all x. Slope fields reveal solution behavior without solving the equation.

Key terms

Sketching a solution: Start at the point and follow the segments.
Horizontal segments: dy/dx = 0.
Equilibrium solution: Constant y where dy/dx = 0 everywhere.
Depends only on x: Segments are constant along vertical lines.
Value: Understand behavior without solving.

Try an AP-style question

Slope field

Segments are horizontal along y = 2, point upward (positive slope) above y = 2, and downward below it. Slopes don't change from left to right.

Which differential equation matches?

dy/dx = y − 2
dy/dx = x − 2
dy/dx = 2 − y
dy/dx = 2y

Show the answer

A. Zero at y = 2, positive for y > 2, depends only on y.

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