Slope fields, separation of variables and exponential models

Learning objectives

Translate a verbal description into a differential equation, for example a rate proportional to the amount present or to the difference between the amount and a fixed limit

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

Sketch a slope field by computing slopes at sample points, sketch a solution curve through a given point, and match a slope field to its differential equation by noting where the slopes are zero, positive, negative or independent of one variable

Solve a separable differential equation by separating variables, integrating both sides and including a single constant, then impose an initial condition to find the particular solution and state the interval on which it is valid

Derive y = y₀e^(kt) as the solution of dy/dt = ky, determine k from given data, and apply the model to growth, radioactive decay and Newton's law of cooling

Use a slope field to describe long-run behavior, identifying equilibrium solutions and whether nearby solutions move toward or away from them