If you throw a rock off a bridge, how far will it fall in the first five seconds? If you are on an airplane from Chicago heading due north and know your latitude at each moment, how can you determine your speed? If you fill a cone with water at a fixed rate, how quickly will the water level rise? Calculus provides a method for answering each of these questions.
Calculus is the study of smoothly varying functions. It uses the central concepts of differentiation and integration to relate how a function changes to the values it takes on.
Before introducing these concepts, we discuss functions, limits, and continuity to clarify what is meant by a function, and what kinds of functions are most appropriate to study. Specifically, we will want the functions we deal with to be continuous, so that they don't jump around haphazardly. Continuity is closely related to the idea of a limit.
After discussing these preliminary ideas, we introduce differentiation, which is the study of the rates of change of functions. After giving the necessary definitions, we introduce a number of practical techniques for computing derivatives. In a series of important applications, we see how the derivative can be used to study the motion of objects, plot graphs of functions, and solve problems of optimization.
Finally we introduce the second major concept in calculus, integration, which is a sort of reverse operation to differentiation, reconstructing information about a quantity from its rate of change. Understanding how the abstract definition of the integral gives rise to a complementary relationship between integration and differentiation is the central theme in this discussion.