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.