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.
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