Download
1 / 7
70 likes | 205 Vues
This guide introduces slope fields, a valuable tool for visualizing solutions to differential equations. Learn how to draw a slope field by determining slopes at various coordinate points and creating small line segments to represent these slopes. By observing the patterns formed, you can infer the family of curves that satisfy the given differential equation. We will also discuss how to sketch a particular solution curve based on initial conditions, such as (1, -2). Mastering slope fields enhances your understanding of differential equations and their solutions.
E N D
7.1 B – Slope Fields • Goal: draw a slope field, and match a slope field to an appropriate equation
Slope Fields Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but recent AP tests have asked students to draw a simple one by hand. • Slope fields can help us produce the family of curves that satisfies a differential equation. • Remember: Differential equations give the slope at any point (x, y), and this information can be used to draw a small piece of the linearization at that point, which approximates the solution curve that passes through that point. This process will be repeated for several points to produce a slope field.
How do we do it? • To make a slope field, plug in coordinate points for x and y and see what the slope is. • Then, draw a tiny line segment with that slope at that point. • You will see patterns allowing you to get an idea of the what the original function is.
Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 0 0 0 0 1 0 0 2 0 0 3 0 2 1 0 1 1 2 2 0 4 -1 -2 0 0 -4 -2
If you know an initial condition, such as (1,-2), you can sketch the particular curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola.
