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Slope Fields quiz #1

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  • What is a slope field, and how does it help visualize solutions to a first-order differential equation such as y' = x - y?
    A slope field, also called a direction field, is a graphical representation that shows the slopes of solutions to a first-order differential equation at various points in the plane. For an equation like y' = x - y, short line segments are drawn at each point (x, y) with a slope equal to x - y. This allows us to see the general shape and behavior of all possible solution curves, even if we cannot solve the equation analytically.
  • What is a slope field (or direction field) in the context of differential equations?
    A slope field is a graphical representation that shows the slopes of solutions to a first-order differential equation at various points in the plane.
  • How do you determine the slope of a line segment at a point (x, y) when drawing a slope field for y' = x - y?
    You plug the coordinates (x, y) into the derivative formula y' = x - y to get the slope at that point.
  • What pattern emerges in the slope field for y' = x - y along the line where x = y?
    Along the line x = y, the slope is zero, so the line segments are horizontal.
  • How can you use a slope field to sketch a particular solution passing through a given point?
    You plot the given point and then draw a curve that follows the direction of the line segments (slopes) in the slope field through that point.
  • Why are slope fields useful for differential equations that are difficult to solve analytically?
    Slope fields allow us to visualize the general behavior and shape of solutions even when we cannot find explicit solutions.
  • What does a positively sloped line segment in a slope field indicate about the solution at that point?
    It indicates that the solution curve is increasing at that point.
  • If you have the point (1, 0) for y' = x - y, what is the slope of the line segment at that point?
    The slope is 1, since 1 - 0 = 1.
  • What happens to the slope field line segments as you move along the diagonal where x - y = 1?
    All the line segments along this diagonal have a slope of 1.
  • When sketching a particular solution on a slope field, why do you only ensure the curve passes through the initial condition?
    Because the initial condition is the only point guaranteed for that solution, and the rest of the curve is guided by the local slopes in the field.