Practice Slope Fields in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
A graphical representation of a first-order DE \frac{dy}{dx} = f(x, y). At each point (x, y) in the plane, draw a short line segment with slope f(x, y). The resulting pattern of segments shows the direction solutions must follow.
Imagine a field with tiny arrows showing which way a river flows at each point. A slope field is the same idea: the DE tells you the slope (direction) at every point, and solution curves are paths that follow these directions everywhere. Drop a 'particle' anywhere and follow the arrows—that's a solution.
Example 1
easyFor dy/dx = y, compute slopes at (0,0), (0,1), (1,1), (1,-1). Describe the solution family.
Example 2
mediumFor dy/dx = x-y, find the zero-slope isocline and describe long-term behavior of solutions.
Example 3
easyFor dy/dx = x^2, compute slopes at (0,0), (1,0), (-1,3), (2,5).
Example 4
mediumA slope field shows solutions curving toward the x-axis from both sides. What can you infer about f(x,y) and long-term behavior?