7.4 Reasoning Using Slope Fields

• An Ordinary Differential Equation (ODE) contains only one independent variable.


  True
• Slope fields can provide insights into the behavior of solution curves without explicitly solving the differential equation.


  True
• Steps to construct a slope field

  1️⃣ Choose a grid of points in the coordinate plane

  2️⃣ At each grid point, draw a short line segment with a slope equal to the derivative at that point

  3️⃣ The collection of these line segments forms the slope field
• What does the slope of a line segment in a slope field represent?

  The value of the derivative
• Line segments in a slope field suggest the solution curve may approach a vertical asymptote
• Critical points in a slope field occur where the slope is zero
• Match the property of a slope field with its interpretation:

  Slope ↔️ Derivative value

  Critical points ↔️ Change in solution direction
• Match the type of differential equation with its example:

  Ordinary Differential Equation (ODE) ↔️ \( y'(x) = x + y(x) \)

  Linear Differential Equation ↔️ \( y'(x) + 3y(x) = 5 \)

  Non-Linear Differential Equation ↔️ \( y'(x) = y^2(x) \)
• Match the slope value with its line segment and direction:

  Positive ↔️ \ (Rising)

  Negative ↔️ / (Falling)

  Zero ↔️ Horizontal (Flat)

  Undefined ↔️ Vertical (Undefined)
• Positive slopes in a slope field indicate solutions that are rising
• By analyzing a slope field, you can infer the general form of the underlying differential
• What do negative slopes in a slope field indicate about the solutions?

  Falling or decreasing solutions
• How can a slope field be used to approximate solutions of a differential equation?

  By sketching solution curves
• The equation \(y'(x) = 2x\) states that the first derivative of \(y(x)\) is equal to 2x
• To construct a slope field, you first choose a grid of points in the coordinate plane.
• A differential equation involves a function and its derivatives
• Match the slope value with its corresponding line segment in a slope field:

  Positive ↔️ \

  Negative ↔️ /

  Zero ↔️ Horizontal

  Undefined ↔️ Vertical
• What does a critical point in a slope field indicate?

  A change in direction
• Steps to match a slope field to a given differential equation

  1️⃣ Examine the slope of the line segments

  2️⃣ Analyze the direction of movement

  3️⃣ Identify critical points

  4️⃣ Look for asymptotic behavior

  5️⃣ Infer the general form of the differential equation
• Critical points in a slope field suggest a change in the direction of the solution curve
• The differential equation \( y'(x) = 2x \) states that the first derivative of \( y(x) \) is equal to 2x
• Slope fields are particularly useful for non-linear and higher-order differential equations


  True
• What does the slope of each line segment in a slope field represent?

  Value of the derivative
• What does asymptotic behavior in a slope field suggest about the solution curve?

  Approaches a vertical asymptote
• Match the property of a slope field with its interpretation:

  Slope ↔️ Value of the derivative

  Critical points ↔️ Change in solution direction

  Asymptotic behavior ↔️ Vertical asymptote
• For the differential equation \( \frac{dy}{dx} = x \), the slope at \( (1, 1) \) is 1
• What is the slope of the solution curve at \(x = -1\) for the differential equation \(y' = x\)?

  -1
• What is a differential equation?

  An equation relating a function to its derivatives
• What is a slope field?

  A visual representation of slopes
• What does a horizontal line segment in a slope field indicate?

  A slope of zero
• What does the orientation and steepness of a line segment in a slope field indicate?

  Direction and magnitude of the derivative
• Positive slopes in a slope field indicate rising or increasing solutions.

  True
• Analyzing a slope field allows insight into the behavior of a solution curve without solving the differential equation

  True
• A slope field is a visual representation of the slope of a differential equation at various points in the coordinate plane


  True
• A differential equation relates a function to its derivatives


  True
• Steps to construct a slope field

  1️⃣ Choose a grid of points

  2️⃣ At each grid point, draw a line segment with the correct slope

  3️⃣ The collection of line segments forms the slope field
• A slope field with horizontal line segments at \( y = 0 \) and increasing steepness as \( y \) moves away from \( 0 \) suggests the differential equation \( y' = y
• Critical points in a slope field occur where the slope is zero.


  True
• A slope field with horizontal line segments at \( y = 0 \) suggests the differential equation \( y' = y \).

  True
• Steps to sketch a slope field

  1️⃣ Choose a grid of points

  2️⃣ Calculate the slope at each point

  3️⃣ Draw line segments with the calculated slope