This question arose
very naturally round about the year 1600 A.D., when all kinds of
moving objects-from planets to
pendulums- were being studied. Men were then just starting to study
the material world intensively. From that study the modern world
has developed, with the knowledge of stars and atoms, of machines
and genes, that we have today, for good and for ill. One might have
expected the study of speed to have very
limited applications-to machinery, to falling objects, to the
movements of the heavenly bodies. But it has not been
so.
Practically every
development in science and mathematics,from 1600 to 1900 A.D., was
connected with calculus. From this single root,in a most unexpected
way, knowledge grew out in all directions.You find calculus applied
to the theory of gravitation, heat, light, sound, electricity,
magnetism; to the flow of water and the design of airplanes.
Calculus enables Maxwell to predict radio twenty years before any
physicist can demonstrate radio experimentally; calculus still
plays a vital role in Einstein's theory of 1916 and in the new
atomic theories of the nineteen twenties. Apart from these, and
many other applications in science, calculus stimulates the
appearance of interesting new branches of pure mathematics. In the
present century, a few branches of mathematics have developed that
do not use calculus. Yet even these are mixed up with subjects
related to calculus. Someone studying these branches without a
background of calculus would be at a terrible disadvantage; he
would meet allusions to calculus; there would be results suggested
by theorems in calculus. No person intending to study mathematics
seriously could possibly leave calculus
out.Calculus, then, is an
indispensable topic, both for the pure and applied mathematician.
And calculus grows from a quite simple idea,the idea of speed.
In the past, people often
thought of calculus as an extremely difficult subject. Then,
particularly in England, teachers began to realize that many things
could be done by calculus in a way that was much simpler and more
interesting than anything in algebra. In English high schools a
student may have two or even three years of calculus. But then some
mathematicians say that this is not good; that calculus is really
more complicated than it appears, and that it should only be taught
by a very well qualified mathematician. Where does the truth lie in
all these conflicting views?A comparison may be
helpful. A lady lives in a quiet village, and every Sunday she
drives herself to church. You ask her if
it is easy to drive a car. "Oh, yes," she says, "I have no
mechanical aptitude, and I find it quite simple." She might find it
less simple if she had to drive in the middle of New York, or take
a heavy truck across the Rockies. But there is no denying the fact;
she can drive a car. And, if she ever did have to drive in heavy
traffic, her experience of handling a car would be of some use to
her. She would not be so helpless as someone who had never driven
at all.The situation in calculus
is somewhat similar. Elementary calculus is like elementary car
driving, not difficult to learn and it enables you to do many
things you could never manage otherwise. But if you wish to push
calculus as far as it will go, you will run into things that are
more complicated.How should calculus be taught then? Should we bother the beginner with warnings that only become important in more advanced work?If we do so, the beginner will be confused because he will not see any need for these warnings. If we do not, we shall be denounced by mathematicians for deceiving the young.
I believe the correct
approach is to do one thing at a time. When you take a student into
a quiet road to drive a car for the first time, he has plenty to do
in learning which is the brake and which the accelerator, how to
steer, and how to park. You do not discuss with him how to deal
with heavy traffic which is not there, nor what he would do if it
were winter and the road were covered with ice. But you might very
well warn him that such conditions exist, so that he does not over
estimate what he knows.Mathematics also is an exploration. As we push out further, we meet new and unexpected situations and we have to revise our ideas. Rules we have used, theorems we have proved turn out to have unforeseen weaknesses. If I were asked to write on a sheet of paper all the statements that I was absolutely sure of, statements that would be true at every time and place, I should leave the paper blank.Source: About Calculus by Saqwyer, Calculus Applications by B.N.Panth, Calculus Course by S.Chari (my teacher)
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