For this visualization, we have detailed instructions for the following software packages: Slopes (from the University of Arizona). Visual Calculus - Slope Fields - 1, problem: Using a slope field, find the graph of a solution of the differential equation with the initial condition y(1)1. Visualization: This graph was generated by, slopes. 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, then the differential equation will be of the form dy / dx constant.
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. For example, the AP Calculus AB and Calculus BC Workshop Packet, Special Topic: Differential Equations and the AP Calculus Professional Development Workshop Materials Special Focus: The Fundamental Theorem of Calculus both contain a slope field program for the TI-83 calculator. Jump to page content, jump to navigation, aP Calculus: Slope Fields 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.
They can also insert slope fields into their documents through a software program such as WinPlot TM Nancy Stephenson teaches at Clements High School in Sugar Land, Texas. She was a member of the AP Calculus Development Committee from 1999 to 2003 and is a College Board consultant.
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.