7.4 Reasoning Using Slope Fields

• Negative slopes in a slope field slant downward from left to right.

  True
• Each short line in a slope field indicates the value of the derivative at that point.


  True
• Positive slopes in a slope field slant upward from left to right.


  True
• What does the direction of a slope field segment indicate?

  The slope's sign
• Match the characteristic of a slope field with its description:

  Slant ↔️ Upward (positive) or downward (negative)

  Length ↔️ Longer or shorter
• Upward-slanting segments in a slope field indicate positive slopes.
• Long line segments in a slope field indicate steeper slopes
• Longer segments in a slope field indicate steeper slopes.


  True
• What type of differential equation would match a slope field with upward-slanting segments of varying lengths?

  Slope increases with x
• What should a solution curve do when it encounters upward-slanting segments in a slope field?

  Curve upward
• What do the line segments in a slope field represent?

  The slope at each point
• Upward-slanting segments in a slope field represent positive slopes, while downward-slanting segments represent negative slopes.zero
• Match the characteristics of slope field segments with their descriptions:

  1️⃣ Direction

  2️⃣ Upward-slanting, downward-slanting, or horizontal

  3️⃣ Length

  4️⃣ Longer segments indicate steeper slopes
• A differential equation relates a function with its derivatives
• The slope of each line segment in a slope field is equal to the value of the derivative
• Match the differential equation with its corresponding slope field characteristic:

  dy/dx = x ↔️ Slopes increase with x

  dy/dx = -y ↔️ Slopes are negative for positive y and positive for negative y

  dy/dx = x - y ↔️ Slopes reflect the difference between x and y
• Each line segment in a slope field represents the slope of the solution curve passing through that point.
• Steps to sketch solution curves using a slope field:

  1️⃣ Identify the direction of the slope field segments

  2️⃣ Observe the length of the segments

  3️⃣ Start with an initial point

  4️⃣ Draw a solution curve that follows the segments
• Horizontal segments in a slope field indicate slopes with a value of zero.
• Each line segment in a slope field indicates the value of the derivative at that point.


  True
• What does a horizontal line segment in a slope field indicate?

  A zero slope
• A slope field is a graphical representation of slopes of a differential
• Components of a slope field

  1️⃣ Coordinates

  2️⃣ Line Segments

  3️⃣ Differential Equation
• Horizontal segments in a slope field indicate a slope of zero
• What does the length of a slope field segment represent?

  Magnitude of the slope
• In a slope field, zero slopes are represented by horizontal segments.


  True
• Longer slope field segments indicate flatter slopes.

  False
• A slope field is a graphical representation of the slopes of a differential equation at various points in the plane
• To apply slope fields to real-world problems, the key is to analyze the direction and length of the slope field segments
• A slope field with a mix of upward-slanting, downward-slanting, and horizontal segments would correspond to a differential equation where the slope changes sign and magnitude across the coordinate plane.


  True
• Horizontal segments in a slope field indicate zero slopes.
• A positive slant in a slope field indicates a positive slope value.
• What type of slopes do downward-slanting segments in a slope field indicate?

  Negative slopes
• What is the primary purpose of a slope field?

  Graphical representation of slopes
• A slope field with upward-slanting segments of varying lengths would match a differential equation where the slope increases with the independent variable.

  True
• Analyzing a slope field can provide insights about the real-world problem being modeled, such as the direction and rate of change
• Steps to sketch a solution curve for the differential equation `dy/dx = x - y`:

  1️⃣ Identify upward-slanting segments for `x > y`

  2️⃣ Draw the curve upward

  3️⃣ Identify downward-slanting segments for `x < y`

  4️⃣ Draw the curve downward
• Match the key components of a slope field with their descriptions:

  Coordinates ↔️ The points in the plane where the slope is evaluated

  Line Segments ↔️ Small lines representing the slope at each point

  Differential Equation ↔️ The equation the slope field approximates
• What does the length of a line segment in a slope field indicate?

  The magnitude of the slope