Direction field (slope field)

Definition

A direction field, or slope field, is a graphical representation of the solutions of a first-order differential equation. It shows the slopes of the solution curves at various points in the plane.

5 Must Know Facts For Your Next Test

1. Direction fields help visualize the behavior of differential equations without finding explicit solutions.
2. Each small segment in a direction field represents the slope derived from the differential equation at that particular point.
3. They are particularly useful for understanding qualitative aspects of differential equations like stability and long-term behavior.
4. Direction fields can be used to sketch approximate solution curves by following the arrows or line segments.
5. The accuracy of sketches using direction fields improves as more segments (or points) are calculated.

Review Questions

Related terms

Differential Equation:

An equation involving derivatives of a function or functions. It describes how one quantity changes with respect to another.

Euler's Method:

A numerical method for solving ordinary differential equations using stepwise linear approximations.

Initial Condition:

$y(x_0) = y_0$ is an example; it specifies the value of the solution at a particular point and is used to find specific solutions to differential equations.