StellarA differential equation is an equation that relates a function to one or more of its derivatives. In AP Calculus, these equations are used to model how a quantity changes rather than just what its value is. For example, instead of being told a position function directly, you may be told how fast position changes with time, or instead of being given a population function, you may be told that its rate of change depends on the population itself. This lesson is about translating verbal descriptions into equations involving derivatives, which is the first step in using differential equations as models.
The key skill is recognizing phrases that describe rates. Words such as “changes at a rate,” “is proportional to,” “increases by,” and “decreases according to” usually signal that a derivative should appear. If a quantity y changes with respect to time t, then its rate of change is often written as dy/dt. If a quantity changes with respect to position x, then may be more appropriate. The goal is not yet to solve the equation, but to build the correct derivative relationship from the context.
The key skill is recognizing phrases that describe rates. Words such as “changes at a rate,” “is proportional to,” “increases by,” and “decreases according to” usually signal that a derivative should appear. If a quantity y changes with respect to time t, then its rate of change is often written as dy/dt. If a quantity changes with respect to position x, then