IB Mathematics AI AHL Slope Fields MAI Study Notes - New Syllabus

A slope field (also known as a direction field) is a graphical tool used to visualize the solutions of a first-order differential equation of the form:

$ \frac{dy}{dx} = f(x, y) $

Instead of solving the differential equation analytically, we sketch small line segments at grid points \( (x, y) \), where each segment has a slope equal to the value of \( f(x, y) \). These segments indicate the direction a solution curve would follow at that point.

Purpose:

• Helps visualize families of solutions without solving the differential equation.
• Gives insight into the behavior of solutions based on initial conditions.
• Used to estimate or sketch particular solutions passing through specific points.

Steps to Interpret a Slope Field:

1. Given a differential equation \( \frac{dy}{dx} = f(x, y) \), evaluate the slope at selected points in the plane.
2. Draw a small line segment with that slope at each point.
3. Sketch possible solution curves that follow the slope directions.

Technology Use:

Many GDCs or software like Desmos, GeoGebra, and TI calculators have the option to generate slope fields from a given differential equation.

Important Notes:

• Solutions to differential equations follow the direction of the slope field.
• Initial conditions determine which particular solution curve is followed.
• Equilibrium solutions can often be identified where slope segments are horizontal (slope = 0).