## Visualizing Solutions

Slope fields are an excellent way to visualize a family of solutions of differential equations. When solving differential equations explicitly, students can use slope fields to verify that the explicit solutions match the graphical solutions. When an explicit solution to a differential equation is not possible, the slope field provides a way to solve the 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 2`x`. Student answers might include `y` = `x`^{2}, `y` = `x`^{2} + 3, `y` = `x`^{2} - 1, and so forth; in other words, `y` = `x`^{2} + `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` = 2`x`. 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` = 2. We pick a starting point on our grid and draw a tiny line segment that passes through our point and that has a slope of 2. When we move to another point, we notice that the slope will also be 2 and that the slope will be 2 no matter what point we consider since `dy`/`dx` = constant.

Next I have the students draw a slope field for the differential equation `dy/dx` = `x` +1. We pick a starting point on our grid and draw a tiny line segment that passes through our point and has the slope that we found. I ask the students to name other points that have the same slope. This time the students should notice that all of the points on the graph that have the same `x`-coordinate 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 draw a slope field for another differential equation, such as `dy`/`dx` = 2`y`. This time the students notice that all of the points that have the same `y`-coordinate have the same slope because this differential equation contains a `y`-term but no `x`-term.

The knowledge the students gain by making these observations helps when I ask them to match a differential equation to a slope field. The students look at the slope field to see if all line segments on the slope field have the same slope: if they do, the differential equation will be of the form `dy`/`dx` = constant. If all line segments in the vertical direction on a slope field have the same slope, then the differential equation does not contain a `y`-term. If all segments in the horizontal direction on a slope field have the same slope, then the differential equation does not contain an `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 we 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 one.

The goal is for students to be able to do the following with slope fields:

- Sketch a slope field for a given differential equation.
- Given a slope field, sketch a solution curve through a given point.
- Match a slope field to a differential equation.
- Match a slope field to a solution of a differential equation.
- Draw conclusions about the solution curves by looking at the slope field.
- Discover any solutions of the form
`y`= constant.

Students should be able to do these types of problems without using a graphing calculator.

Slope fields have been a topic on the AP Calculus BC Exam since 1998 and on the AP Calculus AB Exam since 2004. Teachers can find questions involving slope fields from the AP Exams on the following Released Exams:

- 1998 BC multiple-choice question 24 and free-response question 4
- 2000 BC free-response question 6
- 2002 BC free-response question 5
- 2003 BC multiple-choice question 14
- 2004 AB free-response question 6
- 2004 AB Form B, free-response question 5
- 2005 AB free-response question 6
- 2005 BC free-response question 4
- 2005 AB Form B, free-response question 6
- 2006 BC free-response question 4
- 2006 AB free-response question 5
- 2006 AB Form B, free-response question 5
- 2007 AB Form B, free-response question 5
- 2008 AB free-response question 5
- 2008 BC free-response question 6

All of these free-response questions are available on the AP Calculus BC Exam Information page. Teachers can find additional examples in the *AP Calculus Course and Exam Description* (.pdf/3.72MB).

For calculators that do not have a built-in slope field function, students can add handwritten programs to generate slope fields. For example, the Special Focus materials on Differential Equations (.pdf/662KB)and The Fundamental Theorem of Calculus (.pdf/2.2MB) both contain a slope field program for the TI-83 calculator. Teachers can write their own slope field questions by using their graphing calculators, and can insert the slope fields into a Word document with software such as *TI Connect*^{™} or *WinPlot*^{™}.