Objectives: After working through these materials, the student should be able

• to construct a slope field for a given function;
• to construct the antiderivative of a function using a slope field.

Modules:

• Introduction. A useful tool in the study of indefinite integrals (and differential equations) is a slope field (or direction field). Using slope fields, one can find an antiderivative both graphically and numerically.

Definition. Given a function y = f(x), a slope field is drawn for f by choosing a collection of evenly-spaced points x1, x2, ..., xm along the x-axis and a collection of evenly-spaced points y1, y2, ..., yn along the y-axis. At each point (xi, yj), a small line with slope f(xi) is drawn.

By specifying one point (a, b), we can find an antiderivatve F of f such that b = F(a). If f is continuous then there is only one such antiderivative. By using a slope field, you can get an idea of what the graph of an antiderivative of f looks like. (WHY?)

• A LiveMath notebook to draw slope fields and associated antiderivatives.

• Use the JAVA program Slope Fields Calculator to view a slope field for the function f(x) = sin(x) and to investigate the graphs of various antiderivatives of f.
Instructions for the Slope Fields Calculator are available
• There are programs for computers and graphing calculators that will draw such slope fields for any function f.
• Using the TI-86 graphing calculator to graph slope fields and an antiderivative of a function.
  [using Flash]
• TI-85 graphing calculator programs to graph slope fields.