Introduction:
What is Calculus About?

The
seeds for the main ideas in calculus go back to
ancient times. But calculus itself, in the form we think of it
today, was invented during the 17th century (or, should we say "discovered?" -- I'll stick with "invented"). This invention was part of an explosion of renewed interests and discoveries in physical
science. The crucial early developments are usually credited to the English mathematician Sir
Isaac Newton and the German mathematician Gottfried
Leibniz,
working independently -- but back then, Newton, Leibniz and
their partisans had vigorous and nasty arguments about whose work
had priority. As it developed, caclulus gave scientists a tool for remarkable new insights into how the physical world works, and its creation is considered one of the great accomplishments of the human mind. This sweeping assertion is justified not just by the beauty that mathematicians and many students see in the subject, but also by the fact that calculus still is very useful even centuries after its birth. In fact, its role has become more important than ever as the use of mathematical models has grown beyond traditional areas like physics and engineering into such diverse fields as biology, economics, finance and business. Of course there are also a lot of other other important mathematical tools. Algebra, discrete mathematics, probability and statistics, topology, and computer science all have their role in applications, and each one is a research area in its own right. These 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. As
computing power has developed and become cheaper, many
calculations are possible nowadays that would have been unheard of even
50 years ago. The best calculators can now do complicated graphics and
numeric work, as well as symbolic manipulations. And
computers, running software like Matlab, Mathematica or Maple can do
better still (and produce beautiful output). Technology makes it possible to explore and use calculus in ways that were impractical
even
a few decades ago, but There are lots of details and techniques for us to learn, but if we look at 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 deals with the question: "how fast is some quantity changing?" For example, if you're driving east along a straight highway andIn Calculus I, we develop the ideas of the derivative and the integral and take a look at how they are related. In fact, most of Math 131-132 consists of - developing
the
*informal**concepts*behind ideas such as "rate of change" and "total change" - developing
the
*derivative*and*integral* - seeing
*how derivatives and integrals are connected* - learning
*techniques*for use these ideas efficiently
All of this is hard
work.
But what did you expect in learning about "one of the great accomplishments of the human mind?" Setting up a mathematical model of a complicated real-world situation is often hard, and, except for simple examples, it usually requires detailed knowledge of some other subject such as physics, biology or economics. In a calculus course, everybody can learn the tools and see them applied in some "tidy" or "over-simplified" applications which are manageable for everybody and which hint at the real usefulness of the subject. The biologists, chemists, physicists, engineers, architects, economists, and others who have recommended that you take a calculus course will have to show you the reasons why it's useful in their own fields ( please,
put
them on the spot and ask!! ) For now, try to learn to
appreciate
the subject itself, its beauty, and how the pieces fit together. |