Overview
Given a differential equation \(\displaystyle \frac{dy}{dt} = f(t,y)\), we can evaluate \(f\) at a point \((t,y)\) to find the slope of the tangent (the derivative) of a solution \(y=y(t)\) which passes through this point. Finding the slopes of the tangent line for a wide selection of points in the plane give us a socalled slope field. With this slope field in hand, we can describe the general behavior of solutions of the DE without actually explicitly solving the DE. (An approximative method to solve a DE based on this approach is called "Euler's Method".)
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.

Use a DE to calculate slopes of solutions at given points, and describe these slopes in terms of the physical situation modeled by the DE.

Explain the construction of a slope field. Specifically, explain how slope fields are related to tangent lines, and how a DE can be used to create a slope field.

Graphically sketch a solution to a DE when given all or part of a slope field.

State the meaning of equilibrium solution, and find them for simple DEs.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Construct a slope field by hand.

Give qualitative descriptions of solutions to DE using data given by a slope field.

Calculate and interpret isoclines: The curves for which a DE has a given slope.

Find and describe stable and unstable equilibria.
To prepare for class

Read the Introduction of Section 7.2.

Do the Preview Activity for section 7.2 (on WeBWorK if required by your teacher). You may want to use the electronic version of the textbook since the tangent lines are graphed using color.

Watch the following videos:

Watch the following video which explains how using the slope field of a DE, and an initial condition, we can obtain a graphical representation of a DE.

Read the beginning of Section 7.3.

Do the Preview Activity for section 7.3 (on WeBWorK if required by your teacher).

Do some experimentation with the following interactive applet which allows you to enter a formula and have the corresponding slope field drawn:
Slope Fields (Nestor VallesVillarreal)