Then I hand them a sheet of grid paper and a ruler, and we start with a differential equation such as dy/dx = 2. When I teach my students to draw a slope field, I first review how to graph a line, given a point and a slope. Students can look at the slope field and visualize the family of antiderivatives and can also sketch the solution curve through a particular point. Another way to show the family of antiderivatives is to draw a slope field for dy/dx = 2x. I ask them to sketch several of these antiderivatives on the same graph grid so that they can see the family of antiderivatives. 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. When I introduce antiderivatives to my students, I ask them to name a function whose derivative is 2x.
Slope fields also give us a great way to visualize a family of antiderivatives. When an explicit solution to a differential equation is not possible, the slope field provides a way to solve the equation graphically.
When solving differential equations explicitly, students can use slope fields to verify that the explicit solutions match the graphical solutions. Slope fields are an excellent way to visualize a family of solutions of differential equations.
