7.3 Sketching Slope Fields

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

  Graphical representation of slopes
• A slope field is a graphical representation of the slope of a differential equation at different points in the coordinate plane
• A slope field shows the tangent line at multiple points.
• For the differential equation $\frac{dy}{dx} =$$x$, what is the slope at (1, 1)?


  1
• For $\frac{dy}{dx} =$$x$, the slope field at (1,1) would display a line segment with a slope of 1
• What determines the slope of each line segment in a slope field?

  Differential equation
• For \frac{dy}{dx} = x</latex>, at the point (1, 2), the slope is 1
• How does a slope field visualize the solutions to a differential equation?

  By displaying slopes at points
• The slope field of a differential equation $\frac{dy}{dx} =$$x$ at (1, 1) shows a line segment with a slope of 1.
• Match the features with their corresponding graph type:

  Tangent lines at multiple points ↔️ Slope Field

  Function's shape and behavior ↔️ Typical Graph
• Order the steps to create a slope field for $\frac{dy}{dx} =$$x$


  1️⃣ Calculate the slope at various points

  2️⃣ Draw short line segments with the calculated slopes

  3️⃣ Visualize the qualitative behavior of solutions
• A slope field is created by drawing short line segments at various points, where the slope of each segment is determined by the differential equation
• For the differential equation \frac{dy}{dx} = x</latex>, what is the slope at (2, 3)?

  2
• The slope field of $\frac{dy}{dx} =$$x$ at (0, 0) shows a line segment with a slope of 0.
• What is a slope field used to represent graphically?

  Solutions to a differential equation
• A slope field is created by drawing short line segments where the slope of each segment is determined by the differential equation
• In a slope field, each line segment represents the slope of the tangent line at that point.
• What does a slope field show at multiple points in a graph?

  Tangent lines
• A slope field is used to analyze the shape and behavior of solutions
• Key features of slope fields

  1️⃣ Tangent lines at multiple points

  2️⃣ Understanding solutions to differential equations

  3️⃣ Line segments with varying slopes
• For the differential equation dy/dx = x, at the point (1,2), the slope is 1.
• A slope field is a graphical representation of the solutions to a differential equation
• What is the primary purpose of a slope field in analyzing a differential equation?

  Understanding solutions
• For the differential equation dy/dx = x, at the point (1,2), the slope is 1.
• What are slope lines also known as in the context of slope fields?

  Tangent line segments
• Steps to sketch slope lines

  1️⃣ Choose points on the plane

  2️⃣ Calculate slopes using the differential equation

  3️⃣ Sketch short line segments at each point
• Match the slope type with its visual representation:

  Positive slope ↔️ Line slanting upwards

  Negative slope ↔️ Line slanting downwards

  Zero slope ↔️ Horizontal line

  Undefined slope ↔️ Vertical line
• For dy/dx = x, at the point (0, 1), the slope is 0.
• What is the slope of dy/dx = x at the point (-1, 3)?

  -1
• Steps for calculating and sketching slopes in a slope field

  1️⃣ Determine the slope at each point using the given differential equation $\frac{dy}{dx}$

  2️⃣ Sketch short line segments at each point with the calculated slope
• Match the slope type with its visual representation:

  Positive slope ↔️ Line slanting upwards

  Negative slope ↔️ Line slanting downwards

  Zero slope ↔️ Horizontal line

  Undefined slope ↔️ Vertical line
• For $\frac{dy}{dx} =$$x$, the slope at (0, 1) is 0
• For $\frac{dy}{dx} =$$x$, the slope at (1, 2) is 1
• For $\frac{dy}{dx} =$$x$, the slope at (-1, 3) is -1
• Order the key behaviors of solutions in slope fields:

  1️⃣ Steady States

  2️⃣ Asymptotic Behavior

  3️⃣ Stability
• Match the behavior with its description:

  Steady States ↔️ Solutions where \frac{dy}{dx} = 0</latex>

  Asymptotic Behavior ↔️ Curves converge to a horizontal line

  Stability ↔️ Solutions converge to or diverge from a steady state
• In $\frac{dy}{dx} =$$y(1 - y)$, the steady states are $y =$$0$ and $y =$$< blank_{s}tart > 1 < / blank_{e}nd > < distractors > - 1 ||| 2 ||| undefined < / distractors > < cloze_{e}nd > < truefalse_{s}tart > < line > 43 < / line > < statement_{s}tart > For < latex > \frac{dy}{dx} =$$- y$, solutions approach $y =$$0$ as $x \to \infty$<statement_end><answer_start>True<answer_end><truefalse_end>

  
  

  <cloze_start>A slope field is a graphical representation of the slope of a differential equation
• Each line segment in a slope field represents the tangent line at that point
• Match the feature with its corresponding graph:

  Slope Field ↔️ Tangent lines at multiple points

  Typical Graph ↔️ Function's shape and behavior
• A slope field is created by drawing short line segments at various points, where the slope of each segment is determined by the differential equation