7.4 Reasoning Using Slope Fields

• What does a slope field graph at various points (x, y)?

  The slope
• What are the components of a differential equation?

  Function and derivative
• Why is a slope field useful for understanding differential equations?

  Visualizes solution behavior
• The first step in constructing a slope field involves choosing a grid of points
• What does the resulting slope field diagram show?

  Slopes of solution curves
• Steps to sketch solution curves using a slope field

  1️⃣ Identify the initial condition (x₀, y₀)

  2️⃣ Find the slope at (x₀, y₀) from the slope field

  3️⃣ Draw a short line segment from (x₀, y₀)

  4️⃣ Move a small distance and repeat

  5️⃣ Follow the slopes to sketch the full solution curve
• A differential equation expresses the relationship between a function and its derivatives
• A differential equation expresses the relationship between a function and its derivatives.


  True
• In a slope field, steep lines indicate rapid changes
• Equilibrium solutions occur when `dy/dx = 0
• Equilibrium solutions are constant solutions to a differential equation where `dy/dx = 0
• Match the key concept with its description:

  Equilibrium Solutions ↔️ Constant solutions where `dy/dx = 0`

  Slope Fields ↔️ Show locations of equilibrium solutions
• Asymptotes in a slope field occur when solutions approach but never cross
• What does each line segment in a slope field represent?

  Slope of solution curve
• Different initial conditions in a slope field lead to the same solution curve.

  False
• A slope field allows you to understand the trends of solutions without solving the differential equation explicitly.

  True
• Each line segment in a slope field indicates the slope of the solution curve
• Steps to construct a slope field for a given differential equation

  1️⃣ Choose a grid of points (x, y).

  2️⃣ Calculate the slope dy/dx at each grid point.

  3️⃣ Draw a short line segment at each grid point with the calculated slope.
• Steps to construct a slope field for a differential equation

  1️⃣ Choose a grid of points (x, y)

  2️⃣ Calculate the slope dy/dx at each grid point

  3️⃣ Draw a short line segment at each grid point with the calculated slope
• Once a slope field is constructed, it can be used to sketch the general shape of the solution curves
• Sketching solutions using slope fields allows visualization without solving the differential equation explicitly


  True
• The connection between a differential equation and a slope field is that the slope field visually represents the differential equation


  True
• Match the step with its description for drawing a slope field:

  Step 1 ↔️ Select a grid of points in the domain

  Step 2 ↔️ Calculate the slope at each point

  Step 3 ↔️ Draw a line segment with the calculated slope
• A slope field diagram helps predict the general shape of solution curves.


  True
• Steps for sketching solutions using slope fields:

  1️⃣ Identify the initial condition

  2️⃣ Find the slope at the initial point

  3️⃣ Draw a short line segment

  4️⃣ Move along the segment and repeat

  5️⃣ Sketch the full solution curve
• What is an equilibrium solution for the differential equation `dy/dx = y - 2`?

  y = 2
• Match the key concept with its description:

  Equilibrium Solutions ↔️ Constant solutions where `dy/dx = 0`

  Slope Fields ↔️ Show locations of equilibrium solutions
• For the differential equation `dy/dx = y - 2`, an equilibrium solution is `y = 2
• For the differential equation `dy/dx = y - x`, the slope is zero along the line `y = x`.


  True
• What is a differential equation?

  Function and derivative relationship
• A slope field visualizes the general behavior of solution functions
• The key connection between a differential equation and its slope field is that the field visualizes the equation's solutions
• What do slope fields represent graphically?

  Slopes of solution curves
• A slope field allows you to find the exact solution to a differential equation.

  False
• What is calculated at each grid point when constructing a slope field?

  The slope dy/dx
• A slope field allows you to visualize the general behavior of solution functions without solving the equation explicitly


  True
• A slope field provides a quantitative understanding of the dynamics described by a differential equation

  False
• What does the slope field for the differential equation dy/dx = x show at x = 0?

  Horizontal lines
• What does a slope field visualize?

  General behavior of solutions
• The key connection between differential equations and slope fields is that the slope field provides a visual representation of the differential equation