Slope Fields

AP Calculus · Axiom Academy

Visualizing differential equations without solving them A slope field (or direction field) is a graphical representation of a DE . At each point (x,y) , we draw a short line segment with slope f(x,y) . Solution curves follow the direction of the segments. Without solving the DE algebraically, you can sketch approximate solutions by tracing curves that are tangent to the segments everywhere. At (0,0) : slope = 0 . At (1,0) : slope = 1 . At (0,1) : slope = -1 . At (2,1) : slope = 1 . Draw short segments at each point with the computed slope. Solution curves that start at different initial values will trace paths through the field. Slope fields show the direction that solutions flow Isoclines: curves where f(x,y) = k (constant slope) Solutions never cross (by uniqueness theorem) Practice matching DEs to their slope fields on the AP exam

This is the written version of the interactive lesson above. See the full AP Calculus course.