7.4 Reasoning Using Slope Fields

• What is a differential equation?

  Equation with derivatives
• A slope field is a graphical representation of the solutions to a differential equation
• The slope of the tangent line in a slope field is determined by the differential equation at each point.
• What is the first step in creating a slope field for the differential equation \frac{dy}{dx} = f(x, y)</latex>?

  Choose representative points
• Steps to create a slope field for $\frac{dy}{dx} =$$f(x, y)$ in the correct order


  1️⃣ Choose representative points

  2️⃣ Calculate slopes

  3️⃣ Plot slope segments

  4️⃣ Sketch solution curves
• When creating a slope field, the length of the line segments at each point indicates the magnitude of the slope.

  False
• The final step in creating a slope field is to sketch solution curves
• A differential equation involves a function and its derivatives
• The purpose of a slope field is to model rates of change.

  False
• An example of a differential equation is \frac{dy}{dx} = x + y
• A slope field represents the solutions to a differential equation graphically.
• For the differential equation $\frac{dy}{dx} =$$x +$$y$, the slope field consists of line segments with slopes equal to x + y
• Steps to create a slope field for a differential equation \frac{dy}{dx} = f(x, y)</latex>

  1️⃣ Choose Representative Points

  2️⃣ Calculate Slopes

  3️⃣ Plot Slope Segments

  4️⃣ Sketch Solutions
• Sketching solutions on a slope field involves aligning with the calculated slopes.
• A differential equation is an equation involving an unknown function and its derivatives
• The slope field for \frac{dy}{dx} = x + y</latex> has line segments with slopes equal to $x - y$ at various points.


  False
• Steps to create a slope field for a differential equation $\frac{dy}{dx} =$$f(x, y)$


  1️⃣ Choose Representative Points

  2️⃣ Calculate Slopes

  3️⃣ Plot Slope Segments

  4️⃣ Sketch Solutions
• Calculating slopes in a slope field involves evaluating $f(x, y)$ at each chosen point.
• Use the slope field to sketch solutions
• Match the solution type with its slope field behavior:

  Stable Equilibrium ↔️ Tangent lines converge towards equilibrium

  Unstable Equilibrium ↔️ Tangent lines diverge away from equilibrium

  Asymptotic Approach ↔️ Solutions approach but never reach equilibrium
• The slope field for \frac{dy}{dx} = y(1 - y)</latex> indicates stable equilibrium solutions at $y =$$0$ and $y =$$1$.
• A slope field is a graphical representation of solution slopes
• The purpose of a slope field is to model rates of change.

  False
• To create a slope field, first choose representative points
• After calculating slopes, plot slope segments
• Starting at $(0, 1)$, a solution curve for $\frac{dy}{dx} =$$\frac{1}{2}x - 1$ should align with the local slopes.
• To create a slope field, begin by selecting a grid of points
• What do you calculate at each point in a slope field to determine the slope?

  $f(x, y)$
• After calculating slopes, draw short line segments
• You use a slope field to sketch solutions to the differential equation.
• What is the given differential equation in the example?

  $\frac{dy}{dx} =$$\frac{1}{2}x - 1$
• In the example, the grid of points considered is $(x, y)$
• Steps to create a slope field for a differential equation

  1️⃣ Choose representative points

  2️⃣ Calculate slopes

  3️⃣ Plot slope segments

  4️⃣ Sketch solutions
• Calculating slopes at grid points involves evaluating $f(x, y)$.
• When creating a slope field, short line segments are drawn with the calculated slope
• What is the starting point for sketching the solution curve in the example?

  $(0, 1)$
• To sketch a solution curve, you start at the initial condition point and follow the nearby line segments.
• When sketching a solution curve, adjust your curve to align with the local slopes
• What type of curve results from sketching the solution for $\frac{dy}{dx} =$$x$ with initial condition (0, 0)</latex>?


  Parabola
• Equilibrium solutions occur where $\frac{dy}{dx} =$$0$.