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How do you sketch a slope field for $\frac{dy}{dx} = x + y$ ?

Choose a grid of points (x,y). 2. Calculate the slope (x+y) at each point. 3. Draw a short line segment with that slope at each point.

How do you sketch a slope field for $\frac{dy}{dx} = \frac{x}{y}$ ?

Choose a grid of points (x,y). 2. Calculate the slope (x/y) at each point. Note: slope is undefined when y = 0. 3. Draw a short line segment with that slope at each point.

What is a slope field?

A visual representation of the solutions to a differential equation, showing slopes at various points.

What does a line segment in a slope field represent?

The slope of the solution to the differential equation at that specific point.

What is a differential equation?

An equation that relates a function with its derivatives.

How do slope fields help visualize solutions to differential equations?

By showing the direction a solution curve would take at any given point in the plane.

What information is needed to construct a slope field?

The differential equation $\frac{dy}{dx} = f(x,y)$ and a grid of points in the xy-plane.

How is the slope at a point (x,y) determined in a slope field?

By evaluating the differential equation $\frac{dy}{dx}$ at that point (x,y).

What does a slope of zero indicate in a slope field?

A horizontal line segment, indicating that the solution curve has a local maximum or minimum, or is constant at that point.

What does an undefined slope indicate in a slope field?

A vertical line segment, indicating that the solution curve has a vertical tangent at that point.