1. Collect Homework.
2. Today we are going to study a concept called slope
fields.
Slope fields provide an excellent way to visualize a
family of solutions of differential equations.
Some differential equations can be solved algebraically,
and the slope field for those differential equations can
be used to verify that the algebraic solution matches
the graphical solution.
Other differential equations cannot be solved
algebraically, and the slope field for those
differential equations provides a way to solve the
differential equation graphically.
Slope fields also give us a great way to visualize a
family of antiderivatives.
3. Recall how to graph a line given a point and the slope.
Graph a line passing through (-2, 3) with a slope of 2.
4. Hand out rulers and slope field packets.
Start with a differential equation such as
dy/dx = x + 1.
Pick a starting point on our grid and substitute it into
the differential equation to determine the slope at that
point.
Then we draw a tiny segment that passes through our
point and has the slope that we found.
Next ask the students to name other points that have the
same slope. They notice that all of the points that have
the same x-coordinate will have the same slope because
our differential equation has an x-term but no y-term.
After completing the slope field for dy/dx = x + 1, try
another differential equation, such as dy/dx = 2y.
This time the students notice that all of the points
that have the same y-coordinate will have the same slope
because our differential equation contains a y-term but
no x-term.
This knowledge helps the students when they are asked to
match a differential equation to a slope field.
The student looks at the slope field to see if all of
the segments in the vertical direction have the same
slope; if they do, then the differential equation
contains an x-term but no y-term.
If all of the segments in the horizontal direction on a
slope field have the same slope, then the differential
equation contains a y-term but no x-term.
After making these observations, move on to differential
equations that contain both an x-term and a y-term, such
as dy/dx = x +y and look for points that have the same
slope as we draw the slope field for this differential
equation.
The students like to use a ruler at first to help draw
their segments so that they have the correct slope, but
soon they are able to draw them without using a ruler.
5. Let students work on the slope field packets.