The
seeds for the main
ideas
of calculus go back to ancient times but calculus itself, as we think
of
it today, was invented (or discovered?--which?) during the 17th century
as part of an explosion of interest and discovery in the physical
sciences.
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
gave
scientists a tool to generate remarkable new understandings of the
world.
Its creation is considered one of the great intellectual
accomplishments
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
retains
its fundamental importance, even several centuries after its
birth.
In fact, its role has become more important than ever as the use of
mathematical
models has reached beyond areas such as physics and engineering and
into
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.
Studying rates
of
change
involves a concept from Calculus I called the derivative.
The velocity v is the derivative of the position function s.
If we think of s = f ( t ) as a function of time, then
some
of the ways of writing the derivative are v = f ' ( t ) or v
= ds / dt. A derivative is analogous
to the speedometer on your car.
Of course,
instead
of how
fast s is changing, you might be interested in the rate of
change
of some other quantity. For example,
y
= the
size of a population
of bacteria
y =
the
amount (mass)
of a radioactive isotope in a sample
y =
the
price of a gallon
of gasoline.
Whether a rate of
change occurs
in biology, physics, or economics, the same mathematical concept--the
derivative--is involved in each case.
2)
The other idea is "opposite" to the first one. If you know
how
fast some quantity is changing, then how big is it at a certain
time?
On the highway again, you could imagine trying to figure out how far s
you are from home at time t by studying the velocity v
(speedometer
information).
This is easy to
do
if the
car moves at constant velocity: in that case, s = vt because
distance = (rate)(time). But
finding s
from v is harder if the velocity is variable during the
trip.
The calculation of a total amount s from a rate v
=ds/dt involves a concept from calculus called the integral.
An intergal analogous to your car's trip odometer which (if you
set it to
0
when you start out) measures s on your dashboard.
Instead of a
distance, you
might be interested in trying to compute the total amount of
money
in an account if you know how fast it's growing (the interest rate), or
the total number of people infected with some disease at time t if
you know the infection rate. The same mathematical concept, the
integral, is involved in each case. The study of the
integral
and how it relates to the derivative, is the main theme of Calculus II.
Most of Math
131-132 consists
of
- developing
the background
for these ideas
- developing
the
mathematical
concepts of derivative and integral
- seeing how
the
two ideas
are connected
- learning techniques
for
applying the ideas efficiently.
All of this is hard
work.
But what did you expect in learning about "one of the great
intellectual
accomplishments of the human mind?" The day-to-day work may seem
tedious at times, but it's essential, like finger exercises for the
future
pianist. Or, to change the analogy, it's like learning a new
language:
it can open new vistas but only if you're willing to memorize
vocabulary,
learn to conjugate verbs, and practice, practice, practice!
We'll see
indications
of
some of the diverse applications of calculus during the course. But
it's
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
"more
applications" in math courses often don't realize that applications,
generally,
are much harder: a little like "story problems," only worse. That's
because
applying math to a concrete situation involves taking a complicated,
messy
real-life situation, sorting out what's relevant to the problem and
what
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
mathematical
model of a complicated real-world situation is often not easy, and it
usually
requires detailed knowledge of another subject such as physics, biology
or economics. In a calculus course you learn the tools and see
them
applied in some "tidy" applications which only 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. |