AP® Calculus BC · Unit 7 · Topic 7.3
Sketching Slope Fields
A slope field is a grid of short dashes, each showing the slope dy/dx that a solution curve would have at that exact point, so the whole picture shows every possible solution at once.
A differential equation like dy/dx = x - y gives you a rule for slope at any point (x, y), not just one number. Plug in a point's coordinates and the rule spits out the slope a solution curve must have if it ever passes through there. A slope field draws that slope as a tiny dash at a grid of points across the plane. Do this at enough points and you get a picture of the direction every possible solution is forced to travel, everywhere at once — without solving the equation.
To build one, pick your grid points, plug each (x, y) pair into the right-hand side of the equation, and draw a short segment through that point tilted at the resulting slope. Steep positive slope, the dash leans hard upward left-to-right; zero slope, it's flat; negative, it leans down. No algebra beyond substitution is needed — the equation does the work. Once the grid is full, you can sketch an actual solution curve by starting anywhere and drawing a smooth path that stays tangent to each dash it passes near, like following a trail of arrows.
Slope fields set up everything else in the unit: Euler's method walks along these same dashes numerically, and separable equations give you the formula for the curve the field is only sketching by eye. Where the right-hand side is undefined, such as division by zero, the field simply has no dash, and solution curves there often show vertical tangents or breaks.
Key terms
- Slope field
- A grid of short segments, each showing the slope a solution curve would have at that point.
- Right-hand side
- The expression in x and y that dy/dx equals, used to compute each segment's slope.
- Solution curve
- A smooth curve that stays tangent to the slope field at every point it passes through.
- Integral curve
- Another name for a solution curve, since finding it corresponds to integrating the differential equation.
Watch it explained
- How it worksHow do I compute the slope at each grid point and draw the matching dash?
- Worked exampleDraw the slope field for dy/dx = x - y at the nine grid points where x and y are -1, 0, or 1.
- Common mistakeWhy isn't a slope field segment an actual piece of a solution curve?
- Going furtherWhat happens to a slope field at points where the derivative is undefined?
Check yourself
End of topic.