7.3 Sketching Slope Fields

• A slope field provides insights into the solutions of a differential equation without solving it analytically
• What is the purpose of grid points in a slope field?

  Draw vectors at intervals
• If the slope at a point is positive, the vector points upward to the right
• A slope field provides insights into the solutions of a differential equation without solving it analytically.

  True
• What is typically used to represent the dependent variable in a differential equation?

  y
• Derivatives in a differential equation indicate the rate of change of the dependent variable with respect to the independent variable
• Steps to form a slope field

  1️⃣ Calculate slopes at various points

  2️⃣ Sketch slope vectors at each point

  3️⃣ Connect vectors visually to form a field
• The slope at (1, 1) for the differential equation $\frac{dy}{dx} =$$x - y$ is 0.


  True
• The slope at a point in a slope field is determined by substituting the coordinates into the differential equation.


  True
• When sketching vectors in a slope field, consistency in length is crucial for clarity.


  True
• What do the vectors in a slope field represent?

  The slope at each point
• What is the independent variable typically represented by in a differential equation?

  x
• How do you determine the slope at a specific point in a slope field?

  Substitute coordinates into the equation
• Why are slope fields useful in studying differential equations?

  Visualize solution behavior
• What does a horizontal vector in a slope field indicate?

  Slope is zero
• Steps for sketching slope vectors in a slope field

  1️⃣ Calculate the slope at each point

  2️⃣ Determine the direction based on the slope value

  3️⃣ Sketch the vector aligning with the determined direction
• The axes in a slope field represent the x and y coordinates
• A differential equation relates a function to its derivatives
• The slope at the point (0, 2) for the differential equation $\frac{dy}{dx} =$$x - y$ is -2
• The slope at any point (x, y) for the differential equation $\frac{dy}{dx} =$$2x - y$ is equal to 2x minus y.


  True
• To calculate the slope at a point in a slope field, you substitute the coordinates into the differential equation $\frac{dy}{dx} =$$f(x, y)$.


  True
• A slope field is a visual representation of a differential equation
• Match the key elements of a differential equation with their descriptions:

  Independent variable ↔️ The variable that changes freely

  Dependent variable ↔️ The variable that depends on the independent variable

  Derivatives ↔️ Expressions indicating the rate of change
• What is the dependent variable typically represented by in a differential equation?

  y
• Match the slope value with its corresponding direction:

  m > 0 ↔️ Upward to the right

  m < 0 ↔️ Downward to the right

  m = 0 ↔️ Horizontal

  m undefined ↔️ Vertical
• What does a slope field visually represent?

  Differential equation solutions
• A differential equation relates a function to its derivatives
• What does the value resulting from substituting the coordinates of a point into a differential equation represent in a slope field?

  The slope at that point
• Why are accurate slopes crucial in a slope field?

  To guide understanding of solutions
• What does a horizontal vector in a slope field indicate about the solution curve at that point?

  The solution curve flattens out
• The independent variable changes freely in a differential equation.


  True
• To sketch slope vectors, you must first calculate the slope at each point.


  True
• Match the key components of a slope field with their descriptions:

  Vectors ↔️ Short line segments indicating slope

  Axes ↔️ Represent x and y coordinates

  Grid Points ↔️ Locations where vectors are drawn
• What does a slope of 0 indicate about the direction of the vector in a slope field?

  It is horizontal
• In the differential equation \frac{dy}{dx} = 2x - y</latex>, the derivative is \frac{dy}{dx}
• A slope field visualizes the behavior of solutions to a differential equation.


  True