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 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 it may
produce
an "answer" which is misleading or even completely incorrect!
Technology needs a user who understands and can tell it exactly
what needs to be done. This basic understanding is our goal in a beginning study of calculus.
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 and s = f( t )
represents your distance from starting position at time t, then you might be
interested
in how fast
s is
changing. The rate of change of s at a time t is your velocity v at time t -- which is shown on the dashboard in your car. If s is measured in km and t in hrs, then v has units km/hr.
To study rates
of
change, we use a concept from Calculus I called the derivative. The velocity v, it turns out, is the derivative of the position function s = f (t ). Some
of the ways that people write the velocity (derivative) are v = f ' ( t ) or v
= ds/dt. At any time, the speedometer in your car shows you the current value of the derivative v = ds/dt. When you accelerate, the derivative v gets bigger.
Of course, we might be interested in the rate of
change
of some other quantity. For example, the rate of change of
y
= f( t ) = the
size of a population
of bacteria at time t
y =
f( t ) = the
amount (mass)
of a radioactive isotope present in a sample at time t
V =
f( r ) = the volume of gas in an expanding spherical balloon as the
radius r grows. The
derivative could be written as f ' (r) or as
dV/dr. If the radius is measured in cm and the volume in
cm^3, then the derivative would have units (cm ^ 3)/cm: that
is
(V units)/(r unit)
Whether a quantity is from biology, physics, or economics, the same mathematical tool -- the
derivative -- is what we need to talk mathematically about how fast it changes.
2) The other great idea is "opposite" to the first one: if you know the rate at which some quantity is changing, then by what amount has it changed? On the road again, imagine that you are given all the velocity information v -- that is, all the speedometer
data starting from time 0 when you departed. Then you might like to calculate the amount that s, the distance from your starting position,
has changed after t hours: in other words, how far have you travelled after driving for t hours? (Note:
"distance from the start at time t" could different from "total
distance travelled"-- if, for example, you drove to the east part
of the time and to west part of the time. "Distance from the starting
position" and
"total distance travelled" are the same if you always drive in the same
direction -- always to the east, say.)
This calculation is very easy if your
car moves at constant velocity: for example, if v is constantly 100 km/hr, then we use the simple formula s = (rate)(time) = 100t. But if your velocity varies during the
trip, then finding s from v is harder. Mathematically, we know from part 1) that v = ds/dt and given the values of v; we want to "think backwards" from v to find s. Calculating s(t) - s(0) = the change in position from starting position, from the velocity v
= ds/dt, involves a concept from calculus called the integral. An integral is analogous to your car's odometer
(tripmeter) which tells you distance travelled. If you set it to 0 when you start, then at
any time t, the odometer tells you the total distance travelled at
that moment.
Instead of a
change in distance, you
might be interested in trying to compute the change over one year in the amount of
money
A in an account given the rate of change dA/dt (the interest rate). Or you might want to figure out the change in the total number I of people infected with a disease at time t if
you know the rate of infection dI/dt.
In all these situations, we want to find the "total change" based
on the rate of change. The mathematical tool is the same each
time: the
integral. In
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 exact mathematical meaning of 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?"
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 that can open new vistas and possibilities for you, but only if you're willing to memorize
vocabulary,
learn to conjugate verbs, and practice, practice, practice!
We'll see some ideas illustrating the diverse applications of calculus during the course. But this is
not a course trying to teach physics, biology, economics, or
business. Many of
the most interesting and significant applications you will have to
learn about somewhere else. That should be a relief ! Although it is nice to
get some ideas about what the material is good for, 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 then
are you ready to apply the "tools" you learn in calculus.
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.
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