Objectives: After working through these materials, the student should
- to construct a slope field for a given function;
- to construct the antiderivative of a function using a slope field.
- 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.
- 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.
for the Slope Fields Calculator are available
- There are programs for computers
and graphing calculators that will draw such slope fields for any function