by Nancy Stephenson
Clements High School
Sugar Land, Texas
Visualizing
Solutions
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.
When I introduce antiderivatives to my students, I ask them to name a function
whose derivative is 2x. Student answers might include y = x2,
y = x2 + 3, y = x2 - 1, etc.;
in other words,
y =x2 + C . I ask them to sketch several
of these antiderivatives on the same graph grid so that they can see the family
of antiderivatives. Another way to show the family of antiderivatives is to
draw a slope field for dy/dx = 2x. Students can look at
the slope field and visualize the family of antiderivatives and can also sketch
the solution curve through a particular point.
When I teach my students to draw a slope field, I first review how to graph a
line, given a point and a slope. Then I hand them a sheet of grid paper and a
ruler, and we start with a differential equation such as dy/dx = x
+ 1. We 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 I
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 we complete the slope field for dy/dx = x + 1, we
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, we 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.
Students should be able to do the following with slope fields:
1. Sketch a slope field
for a given differential equation;
2. Given a slope field,
sketch a solution curve through a given point;
3. Match a slope field to
a differential equation;
4. Match a slope field to
a solution of a differential equation.
Students
should be able to do these types of problems without a graphing calculator.
Slope fields have been a topic on the AP Calculus BC Exam since 1998. Questions
involving slope fields can be found on the following exams:
An
additional example can be found in the AP Calculus Course Description in the
Sample Multiple-Choice Questions for Calculus BC, question 6. [Note: all the
AP Calculus free-response questions starting with 1998 are available on AP
Central.]
Some graphing calculators, such as the TI-86 and TI-89, have the ability to
draw slope fields as a built-in capability. For other calculators which do not
have slope fields built in, programs can be written to generate slope fields.
Teachers can write their own slope field questions by using their graphing
calculator and can insert the slope fields into a word-processing document by
using technology such as the TI-GraphLink or TI-Connect. This can also be done
by using computer software programs for slope fields.
Nancy Stephenson teaches at Clements High School in Sugar Land, Texas. She
is a member of the AP Calculus Development Committee.