7.3 Sketching Slope Fields

• What does a slope field visually represent?

  The slope of a differential equation
• A solution curve provides a quantitative solution to a differential equation.


  True
• What can you infer from a slope field without solving the differential equation?

  Direction and curvature
• A slope field shows the slope at each point
• The order of a differential equation is determined by the highest order derivative
• A slope field provides a quantitative solution to a differential equation.

  False
• A slope field shows the slope at each point
• What is a key example of a variable in a differential equation?

  `y` depends on `x`
• A 3x3 coordinate grid is an example of a high-level detail grid.

  False
• Steps to select points on a grid for a slope field:

  1️⃣ Establish the grid coordinates

  2️⃣ Define the grid range

  3️⃣ Select intersections of axes
• What are grid coordinates in the context of slope fields?

  (x, y) pairs
• The base of the slope field is formed by the intersections of the x-axis and y-axis on the grid
• Steps to create a slope field

  1️⃣ Identify the differential equation

  2️⃣ Evaluate the slope at each grid point

  3️⃣ Draw a short line segment at each point
• Match the concept with its description:

  Slope Field ↔️ Shows the slope at each point

  Solution Curve ↔️ Shows the actual solution
• A coordinate grid for a slope field typically has evenly spaced x-axis and y-axis


  True
• Match the characteristic of a differential equation with its description:

  Variables ↔️ Includes dependent and independent variables

  Derivatives ↔️ Contains derivatives of the function

  Order ↔️ The highest order derivative present
• `d²y/dx² + 3(dy/dx) + 2y = sin(x)` is a second-order linear differential equation.


  True
• Steps to create a slope field:

  1️⃣ Create a coordinate grid

  2️⃣ Select points on the grid

  3️⃣ Evaluate the slope at each selected point

  4️⃣ Draw short line segments representing the slopes

  5️⃣ Connect the line segments to create the slope field
• A coordinate grid for a slope field typically has evenly spaced axes
• Why is a coordinate grid essential for creating a slope field?

  To visualize slopes
• Match the feature of a coordinate grid with its description:

  Axes ↔️ Two perpendicular axes, `x` and `y`

  Spacing ↔️ Even spacing between grid points

  Grid Points ↔️ Points where the axes intersect
• What type of grid is typically used in a coordinate grid for creating slope fields?

  Square
• The coordinate grid provides the structure to accurately depict the slope
• Where are the points selected on the coordinate grid to create a slope field?

  Intersections of axes
• What does the grid range determine in a slope field?

  Extent of the plane
• What do the (x, y) pairs on a grid represent in a slope field?

  Grid coordinates
• The grid coordinates in a slope field represent the (x, y) pairs where slopes are computed


  True
• The range of values on the axes is known as the **grid range
• A slope field provides a qualitative understanding of a differential equation without solving it

  True
• A slope field shows the slope of a differential equation at different points in the plane
• The order of a differential equation is determined by the highest order derivative present


  True
• An example of a second-order linear differential equation is `d²y/dx² + 3(dy/dx) + 2y = sin(x)
• Arrange the grid sizes by increasing level of detail

  1️⃣ 3x3

  2️⃣ 5x5

  3️⃣ 7x7
• To create a slope field, you need to select points on the coordinate grid where the axes intersect
• The (x, y) pairs where slopes are computed are called grid coordinates
• Match the grid element with its description:

  Grid Coordinates ↔️ (x, y) pairs for slope computation

  Grid Range ↔️ Extent of the coordinate plane
• What are the grid points on a coordinate grid for creating a slope field?

  (x, y) pairs
• Evaluating the slope at each grid point involves substituting the coordinates into the differential equation.


  True
• What determines the direction of the short line segments drawn in a slope field?

  Slope value
• Match the slope value with its corresponding line segment direction:

  Positive Slope ↔️ Slants upward from left to right

  Negative Slope ↔️ Slants downward from left to right

  Zero Slope ↔️ Horizontal line