7.4 Reasoning Using Slope Fields

• What is a slope field in the context of differential equations?

  Graphical representation of solutions
• The line segments in a slope field depict the instantaneous rate of change defined by the differential equation.


  True
• The second step in constructing a slope field is to calculate the slope at each point
• By tracing the line segments in a slope field, you can visualize possible solutions to the differential equation.


  True
• Consider the differential equation $\frac{dy}{dx} =$$x - y$. At $(0, 0)$, the slope $\frac{dy}{dx} =$ 0
• What does a horizontal slope at $y =$$k$ in a slope field indicate about the differential equation?


  f(k) = 0</latex>
• What is the slope along the line $y =$$x$ for the differential equation $\frac{dy}{dx} =$$x - y$?


  0
• What does a short line segment in a slope field represent at each point $(x, y)$?


  Instantaneous rate of change
• What can you visualize by interpreting a slope field?

  Solution curves
• What does a horizontal slope at $y =$$k$ in a slope field imply about the equation $\frac{dy}{dx} =$$f(y)$?


  $f(k) =$$0$
• Vertical slopes occur when the rate of change is undefined.


  True
• What slope field feature indicates the rate of change is zero at specific y-values?

  Horizontal slopes
• A positive slope in Quadrant I means the rate of change is positive when both x and y are positive.


  True
• For the differential equation $\frac{dy}{dx} =$$x - y$, the slopes are zero along the line y = x
• Steps to draw a slope field for a given differential equation

  1️⃣ Choose points in the xy-plane

  2️⃣ Calculate the slope at each point

  3️⃣ Draw a short line segment with the calculated slope
• A slope field can help identify equilibrium solutions where the slope is zero.


  True
• Equilibrium solutions in a slope field are represented by horizontal line segments where the slope is zero
• Visualizing solutions to a differential equation is possible by plotting line segments in a slope field.


  True
• What is one use of a slope field?

  Visualize solution curves
• Horizontal slopes in a slope field indicate equilibrium solutions where \frac{dy}{dx} = 0</latex>
• Match the key slope field features with their implications:

  Horizontal slopes ↔️ Equilibrium solutions

  Arrows converging to a line ↔️ Stable equilibrium

  Arrows diverging from a line ↔️ Unstable equilibrium
• Steps to construct a slope field

  1️⃣ Choose points to sample across the xy-plane

  2️⃣ Calculate the slope at each point

  3️⃣ Draw a short line segment with the calculated slope
• Horizontal slopes at y = k in a slope field indicate f(y) with f(k) = 0.


  True
• What is the slope of the differential equation $\frac{dy}{dx} =$$x - y$ along the line $y =$$x$?


  Zero
• The slope of the differential equation $\frac{dy}{dx} =$$x - y$ is negative in the upper right quadrant.


  True
• Which term in the equation $\frac{dy}{dx} =$$x - y$ determines the features of the slope field?


  $x - y$
• When is the slope of the differential equation positive in Quadrant I?

  $x > 0$ and y > 0</latex>
• Steps to draw a slope field for a first-order differential equation

  1️⃣ Choose points in the xy-plane

  2️⃣ Calculate the slope at each point

  3️⃣ Draw a short line segment with the calculated slope
• What does it mean if solutions near a certain point are stable in a slope field?

  Solutions approach the point
• The line segment at the point $(1, 1)$ for the differential equation $\frac{dy}{dx} =$$x - y$ is horizontal.
• Match the slope field feature with its implication for solution behavior:

  Horizontal slopes ↔️ Equilibrium solutions

  Arrows converging to a line ↔️ Stable equilibrium solutions

  Arrows diverging from a line ↔️ Unstable equilibrium solutions
• What type of equilibrium solution does the line $y =$$x$ represent for the differential equation $\frac{dy}{dx} =$$y - x$?


  Stable
• At what value of $y$ does the slope field for $\frac{dy}{dx} =$$y - 1$ have horizontal lines?


  $y =$$1$
• Equilibrium solutions in a slope field occur where $\frac{dy}{dx} =$$0$ and are represented by horizontal lines
• For the differential equation $\frac{dy}{dx} =$$y - 1$, at what value of $y$ does the slope field have horizontal lines?


  y = 1
• In a stable equilibrium solution, solution curves near the equilibrium will approach it.
• For the differential equation $\frac{dy}{dx} =$$x - y$, what type of equilibrium solution occurs at $y =$$x$?


  Stable equilibrium
• What happens to solutions when $y > x$ for the differential equation $\frac{dy}{dx} =$$x - y$?


  Solutions increase
• A slope field assigns a short line segment to each point in the xy-plane, indicating the slope of the solution curve passing through that point
• What is the first step in constructing a slope field?

  Choose points to sample