Slope Fields Worksheet
Explore slope fields, their construction, and their relationship to differential equations with this Grade 12 Calculus worksheet.
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Slope Fields Exploration
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Read each question carefully and provide your answers in the space provided. Show all your work for full credit.
1. What is a slope field (or direction field) and what is its purpose in the study of differential equations?
2. For the differential equation dy/dx = x + y, complete the table below to find the slope at the given points.
(x, y)
dy/dx = x + y
Slope
(0, 0)
(1, 0)
(0, 1)
(1, 1)
3. On the coordinate plane below, sketch a small segment of the slope field for dy/dx = y at the points (-2,1), (-1,1), (0,1), (1,1), (2,1) and (-2,-1), (-1,-1), (0,-1), (1,-1), (2,-1).
4. Which of the following differential equations best represents the slope field shown below?
dy/dx = x
dy/dx = y
dy/dx = -x
dy/dx = -y
5. A slope field can be used to visualize the general behavior of solutions to a differential equation without actually solving it.
True
False
6. Given a point (x₀, y₀) on a slope field, how can you sketch a particular solution curve for the differential equation?
7. Match the following differential equations with their corresponding slope fields (A, B, or C).
a. dy/dx = x
b. dy/dx = y
c. dy/dx = -x
8. What is an autonomous differential equation? How does its slope field differ from a non-autonomous differential equation?