7.3 Sketching Slope Fields

• What is a slope field?

  Visual representation of slopes
• Slope fields visualize the behavior of solutions, while explicit solutions provide the exact solution to the differential equation.
• The relationship between the slope and the coordinates in a slope field defines the differential equation.


  True
• Steps to identify the differential equation represented by a slope field

  1️⃣ Observe the direction and magnitude of the slope arrows

  2️⃣ Look for relationships between the slope and the coordinates

  3️⃣ Infer the general form of the differential equation
• The slope field for the differential equation $\frac{dy}{dx} =$$y$ shows that the slopes increase with y
• What is the first step in sketching a slope field?

  Choose representative points
• What does a slope arrow indicate in a slope field?

  Direction and magnitude of slope
• What should the length of line segments in a slope field be?

  Uniform across the field
• What should you look for in a slope field to identify the corresponding differential equation?

  Patterns in slope arrows
• To calculate the slope at a representative point, substitute its coordinates into the differential equation
• What type of behavior do solutions for <tex>\frac{dy}{dx} = x - y</tex> exhibit in its slope field?

  Linear
• Equilibrium points in a slope field can be stable or unstable.


  True
• When sketching a slope field, the length of the slope arrows indicates the magnitude of the slope.


  True
• Steps to evaluate the slope at representative points

  1️⃣ Choose representative points across the plane

  2️⃣ Substitute the coordinates into the differential equation

  3️⃣ Calculate the slope at each point

  4️⃣ Draw slope arrows indicating direction and magnitude
• Match the characteristic with the method:

  Slope Fields ↔️ Visualize the behavior of solutions

  Explicit Solutions ↔️ Provide the exact solution
• What coordinates are typically chosen for representative points in a slope field?

  Integers
• Representative points for sketching a slope field are typically chosen at integer coordinates.


  True
• The length of line segments in a slope field should be uniform.

  True
• What are equilibrium points in a slope field?

  Points where the slope is zero
• Slope fields are useful for visualizing the behavior of solutions to differential equations that cannot be solved explicitly.


  True
• What are the two primary purposes of a slope field?

  Visualize and estimate solutions
• Match the differential equation with its slope field characteristics:

  $\frac{dy}{dx} =$$x$ ↔️ Parallel vertical lines

  $\frac{dy}{dx} =$$y$ ↔️ Parallel horizontal lines

  $\frac{dy}{dx} =$$x +$$y$ ↔️ Diagonally changing direction

  $\frac{dy}{dx} =$$xy$ ↔️ Curved pattern, zero along axes
• To calculate the slope at a representative point, substitute its x and y coordinates into the differential equation.
• To sketch a slope field, it's essential to evaluate the slope at various representative points
• A slope field can visualize solutions even when the differential equation cannot be solved explicitly.

  True
• When analyzing a slope field, it is important to identify points where the slopes are zero
• Steps to sketch a slope field

  1️⃣ Choose representative points across the plane

  2️⃣ Substitute the coordinates into the differential equation to calculate the slope

  3️⃣ Draw slope arrows at each point indicating direction and magnitude
• To draw a line segment in a slope field, orient it to match the calculated slope
• Match the solution type with its characteristic:

  Explicit solution ↔️ Provides the mathematical form of solutions

  Slope field ↔️ Provides a visual overview of solutions
• Individual line segments in a slope field depict the slope at specific points