seeds for the main
of calculus go back to ancient times but calculus itself, as we think
it today, was invented (or discovered?--which?) during the 17th century
as part of an explosion of interest and discovery in the physical
Its invention is usually attributed to the English mathematician Sir
Isaac Newton and the German mathematician Gottfried
Leibniz, working independently. As its power developed, calculus
scientists a tool to generate remarkable new understandings of the
Its creation is considered one of the great intellectual
of the human mind. This sweeping assertion is justified not only
by the beauty of the subject, but also by the fact that it still
its fundamental importance, even several centuries after its
In fact, its role has become more important than ever as the use of
models has reached beyond areas such as physics and engineering and
such different fields as biology, economics and business.
Of course there are also other important mathematical tools. Algebra, discrete mathematics, probability and statistics, topology, and computer science all have roles to play as research areas and as tools in applications. These diverse parts of mathematics complement each other. The increasing power and availability of technology enhances their usefulness and doesn't replace the need for any of them.
Graphing calculators are now good enough to be really helpful with numeric calculations and graphical interpretations of what's happening in calculus. The more powerful calculators (such as the TI-89, TI-92, HP-48 and HP-49) contain a Computer Algebra System (CAS) that can do complex symbolic manipulations. Computers, of course, can do even better and with much prettier output. Technology makes it possible to explore calculus numerically and graphically in ways which were impractical even a decade ago, but technology cannot replace understanding the subject. A calculator or computer is only an assistant that needs an intelligent user. Otherwise it may be unable to find an answer, or may produce an "answer" which is misleading or even completely incorrect ! The technology needs a user who understands and can tell it exactly what it's supposed to do.
There are lots of details and techniques for us to learn, but in the big picture there are only two "great ideas" in calculus. Both of them are illustrated on the dashboard of your car.
1) The first idea concerns "how fast is a quantity changing?" For example, if you're driving down the highway and s represents the distance you've traveled from home, then you might be interested in how fast s is changing (measured, perhaps, in km/hr). How fast s is changing at a time t is your velocity v at that time.Most of Math 131-132 consists of
some of the diverse applications of calculus during the course. But
not a course in physics, biology, economics, or business. Many of
the most interesting and significant applications you will have to meet
elsewhere. That should be a relief! It's certainly nice to
get ideas about what the material is good for, but students who want
applications" in math courses often don't realize that applications,
are much harder: a little like "story problems," only worse. That's
applying math to a concrete situation involves taking a complicated,
real-life situation, sorting out what's relevant to the problem and
isn't, creating a mathematical approximation ("model") to reality, and
then setting up a mathematical formulation of the problem. Only at that
point are you ready to apply the "tools" from calculus. Setting up a
model of a complicated real-world situation is often not easy, and it
requires detailed knowledge of another subject such as physics, biology
or economics. In a calculus course you learn the tools and see
applied in some "tidy" applications which only hint at the real
of the subject. The biologists, chemists, physicists, engineers,
economists, and others who have recommended that you take a calculus
will have to show you the reasons why it's useful in their own fields (please,
them on the spot and ask!! ) For now, try to learn to
the subject itself, its beauty, and how the pieces fit together.