Modeling with differential equations and slope fields

Learning objectives

Write a differential equation from a verbal description, such as a rate of change proportional to the quantity itself or to the difference between the quantity and a constant

Verify that a proposed function is a solution of a differential equation by substituting it and its derivatives into the equation, and find the constants that make it a solution

Sketch a slope field for a first-order differential equation by computing the slope at given points, and sketch a solution curve through a given point on a slope field

Match a slope field to its differential equation by reasoning about where the slopes are zero, positive or negative, and whether they depend only on x, only on y or on both

Use a slope field to describe the behavior of solutions, including whether they increase or decrease, approach an equilibrium value or change concavity