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:

• 1998 BC Multiple-Choice question 24
• 1998 BC Free-Response question 4
• 2000 BC Free-Response question 6
• 2002 BC Free Response question 5

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.