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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.
How do you sketch a slope field for $\frac{dy}{dx} = x + y$?
1. 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 $\frac{dy}{dx} = \frac{x}{y}$?
1. 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.