AB 7.4- Slope Fields

Front

What is a slope field?

Back

A slope field is a graphical representation of the solutions to a first-order differential equation. It consists of a grid of points where each point has a small line segment indicating the slope of the solution curve at that point.

Front

What does dy/dx represent in a slope field?

Back

dy/dx represents the derivative of y with respect to x, indicating the slope of the tangent line to the solution curve at any given point.

Front

How do you determine the slope at a point in a slope field?

Back

To determine the slope at a point, substitute the x and y coordinates of the point into the differential equation dy/dx.

Front

What is the significance of the slope in a slope field?

Back

The slope indicates the direction in which the solution curve is moving at that point. It helps visualize the behavior of the solutions.

Front

What is the general form of a first-order differential equation?

Back

The general form is dy/dx = f(x, y), where f is a function of x and y.

Front

What does it mean if the slope field is horizontal at a point?

Back

If the slope field is horizontal at a point, it means that the derivative dy/dx is zero at that point, indicating a potential equilibrium solution.

Front

What is the relationship between slope fields and solution curves?

Back

Slope fields provide a visual representation of the slopes of solution curves, allowing us to sketch the curves without solving the differential equation explicitly.

Front

How can you identify a particular solution from a slope field?

Back

A particular solution can be identified by finding a curve that passes through a specified point and follows the slopes indicated by the slope field.

Front

What is the derivative of the function y = x^2?

Back

The derivative is dy/dx = 2x.

Front

What does the slope field dy/dx = 2x look like?

Back

The slope field will show increasing slopes as x increases, indicating that the solution curves are concave up.