Introduction to Calculus
Chapter 1 - Introduction
Section 1.1 - Why Study Calculus
As a student you probably view calculus as another
illogical memorization of equations that one needs to pass through school.
You feel you will never be called upon to use any of the 'hypothetical
gibberish' that you learn. Consequently, you approach the subject with
philosophical indifference and adapt yourself to endure that which you can
Perhaps somewhere behind this ill-sentiment is the belief that calculus is
a necessary tool for your existence in the real-world. Regardless of how
one defines a successful existence, an understanding of calculus is
not essential to your well-being. In fact it is an overstatement to
claim that one needs it to become even a scientist.
After all, many
professional engineers do not hesitate to admit their ignorance of the
subject. Such engineer's understanding of calculus is limited to knowing
the equations and knowing how to use them.
If an engineer has a good feel for what to expect based on experience,
then number crunching is not so important. The
best engineers only need the number crunching to confirm what they think the result may be.
So far this introduction has only succeeded in confirming what you may
already feel about your study of calculus, i.e it's a meaningless waste of
time! Why do you have to be bothered learning it while others get by with absolutely
no understanding of it? What more can there be to calculus apart from memorization,
manipulation and frustration? Without fueling your growing suspicions any further, let
us return to the original question and answer it from a purely non-materialistic point of
Why study calculus? This question is best answered with another question; What is
calculus about? It is extremely difficult to answer this briefly and convincingly for that
would require paraphrasing this entire book into just a few lines of text. I could present
you with a series of situations where calculus is applied but that would give you the false
impression that practical applications and mathematical theory are linked together by chance
alone. The simplest definition I can offer is that calculus is the study of mathematically
defined change. There are two words in that definition that require further explanation
before it can make sense. These words are mathematics and change .
What is mathematics? Essentially mathematics is nothing more than the language of
science. While science is a systematic study of nature, mathematics is a concise form of
communication used to represent nature. Man's faculty of reason allow him to observe,
dissect and hypothesize nature and all its ongoing processes such that the end result of
this orderly analysis is mathematics. But what is meant by science and nature? Rather than
continue with questions answered only by generalizations, let us begin this study of
calculus with a study of Man.
What makes man act, think, feel, move and function? The subject of
philosophy is devoted entirely to answering questions like these, though
it often asks more than it is prepared to answer. Life around us
constantly challenges us to live and respond to the seemingly impossible
and unfathomable. We are confronted with dilemmas that require our
judgment. However, the code of ethics and
morals we live by turn out to be nothing more than a set of ironic
contradictions. We find ourselves surrounded by a sea of chaos.
Life then carries us through this sea of
confusion, never stopping anywhere to drop us at a destination.
Life appears futile with no meaning in sight.
Is man confined to this helpless state of existence? To be honest,
no one knows the answer. We can, however, look for hope in the mind. The
power of thought, reason and logic are what allow man to seize control
of life and ultimately serve as a guide to enlightenment and
wisdom. Aristotle felt that happiness could only be achieved by
cultivating the mind. He wrote:
Now the peculiar excellence of man is his power of thought; it is by this faculty that
he surpasses and rules other forms of life; and as the growth of this faculty has given him
supremacy, so, we may presume, its development will give him fulfillment and happiness.
This leads to the question, How do we define reason?. This is synonymous with
asking how do thoughts define behavior? As a child you had little control over your
actions. There did not seem to be any purpose behind what you did or what happened to you.
Life carried you along, giving you the freedom to enjoy the ride.
The idea of free-will offers a more plausible explanation of human actions. It says that man
has the freedom of choice to decide on a certain course of action. Actions are the
results of choices; choices which we are free to make.
The branch of philosophy known as determinism takes free-will a step further to explain
why man chooses to act in one way over another. When man is confronted with a situation,
the line of action he decided on taking is based on experience, personal
preferences. The decisions we make are thus entirely influenced by our past experiences. The
will is not free to behave on its own. For example my decision to write this book
was not based on some impulsive instinct but on a culmination of circumstances I was exposed
to and reacted against.
Any action from the most random to the most perverse can be
explained by the set of situations, experiences, thoughts and feelings
that preceded that action. Essentially determinism says that our lives
are predetermined to the extent that regardless of how we live we can
never change our course of life. As a French thinker once said
, " We change yet we remain the same ". The logic behind this
statement is that if our past can never be changed then our future will
always remain the same. Each action is dictated by the previous one.
It is at this point where science and art diverge. For the scientist, determinism is an
accurate enough explanation of human life, as it says that everything that occurs, occurs
for a set of reasons. Observing and understanding these set of reasons becomes the work of
the scientist. The artist, however, interprets determinism as saying, 'Since everything in
life is predetermined then life is meaningless'.
Life may have no meaning but it is the goal of the artist to question
that statement by exploring the mysterious depths of human nature and
the heart. Perhaps it is the utter randomness of life that causes us to
ignore our unchanging destiny. No human being has any control over
actions that he or she will be exposed to. Fate begins to lose meaning
as one never knows what will happen to oneself. It takes the risks
of the artist to create chaos from which wisdom evolves.
On the other hand, the study of nature is more precise and is less
likely to be influenced by a wide variety of unrelated factors. Within
natures all actions, occurrences, or changes are dependent on a few
factors that can be carefully isolated and studied individually.
Science is specifically about analyzing these interacting systems and
then forming hypothesis that can accurately explain them.
What makes observing phenomena in nature so interesting is that they always occur in a
closed setting where external factors can easily be removed to leave behind just a few
interacting objects. It is these objects along with their properties, that become the focus
of study. Any attempt to logically explain their unique interaction must come from the
objects themselves and not from imaginary external factors. Through reasoning and
observation, nature can be understood, such that the future can be determined from the
present. As Sherlock Holmes would often warn Watson, " You see but you do not observe!"
Often enough, human beings fail to grasp this simple rule of nature by ignorantly
attributing any naturally occurring phenomena to the Gods, heavens, or some mysterious
substance with superpowers. Understanding and accepting the truth requires an open and
critical mind. Charlotte Bronte humorously wrote about this ironic flaw of human nature in
her popular novel, Shirley.
Note well! Whenever you present the actual, simple, truth, it is somehow
always denounced as a lie: they disown it, cast it off, throw it on the
parish; whereas the product of your own imagination, the mere figment, the sheer
fiction, is adapted, termed pretty, proper, sweetly natural: the little spurious wretch gets
all the comfits - the honest, lawful bantling all the cuffs. Such is the way of the
Not only is science stymied by ignorance and deception, but it is also
muddled by the work of the pseudo-scientist. The pseudo-scientist is
described by the Spanish philosopher, Jose Ortega, in his stunning book
on modern western civilization, The Revolt of the Masses:
"... By 1890 a third generation takes command in the
intellectual world, and we find a type of scientist without precedent in
history. He is a person who knows, of all that a routinely dutiful man
must know, only something of one specific science; even of this science,
he is well informed only within that limited area in which he is an
active researcher. He may even go so far as to claim he has an advantage in
not cultivating what lies outside his own narrow field, and he may
declare that curiosity about general knowledge is the sign of the
amateur, the dilettante.
Immured within his small area, he succeeds in discovering
new facts, advances the science which he scarcely knows, and increases
perforce the encyclopedia of knowledge of which he is conscientiously
Physics Nobel laureate Richard Feynman wrote the following observations
about the different types of pseudo-scientists he met at a
conference. Most of the scientists' work
fell into one of these categories:
- Completely ununderstandable
- Vague and indefinite
- Something correct that is obvious and self-evident, worked out by
a long and difficult analysis and presented as an important discovery
- A claim based on the stupidity of the author that some obvious
and correct thing accepted and checked for years is in fact false
- An attempt to do something probably impossible but of certainly
no utility which it is finally revealed at the end fails
- Just plain wrong
One of the remarkable masterpieces of the mind is the science of
mathematics, often called the science of deductive reasoning. While science
is a logical system of thought used to study the natural world;
mathematics is the precise language of science. It is the form
of communication for scientific analysis. Number and symbols are nothing
more than vague abstractions unless they refer to something specific.
Before mathematics can exist there must be a situation to give it
meaning. It is scientific analysis that determines the structure of
Through mathematics we are able to define the present. The present is only dependent on the
conditions that exist within the short frame of time that it occupies. Quickly it vanishes
before our eyes, becoming a memory. The goal of science is to define the objective world in
terms of existing quantifiable conditions expressed by mathematics. Our dimensions or
properties remain fixed and do not change.
It is when our dimensions change that our study becomes a bit more complicated and
calculus arises. But first, what is meant by change? To understand change we need to explain
the concept of time. By definition, time is a passage of events, such that for time to pass,
something must change with respect to itself. For example a moving object implies a
changing distance covered from a reference point. This comprises an event that defines
time. Or a rising temperature implies that the temperature is changing , thus occupying
time. Changes are the results of actions that comprise a situation.
While calculus is the study of mathematically defined change,
it is not necessarily the
study of time alone. In science other dimensions can be changing with respect to each other.
For example velocity can change with height, temperature change with energy, density change
with depth, force change with mass etc.
When a dimension is changing with respect to itself, we say it is changing with respect to
time. When factors change with respect to each other, we disregard the effect time has on
the factors and proceed to only analyze the interacting dimensions. We assume our factors
are constant that change with respect to each other, not with time. Calculus is thus the
branch of mathematics used to study any phenomena involving change . Change is a
relative concept that can involve any pair of dimensions, time, force, mass, length,
temperature etc. This may sound a bit abstract but it will become much clearer as we follow
through the course.
This concludes the answer to what calculus is about. You may not feel I answered the
original posted question, " Why study calculus?". The purpose of studying calculus is simply
to introduce your mind to the scientific method of analysis. Through science, practical
problems can be identified, explanations generated and logical solutions selected. The aim
is for you to understand how to apply your mind in a systematic manner toward understanding
the world around you.
Engineering relies more on this fundamental logical approach toward problem solving
than it does on sheer number crunching and formula manipulation. For this reason many
engineers have forgotten all the theorems of calculus, but what remains is the important
conceptual framework of method and application.
Engineers often jokingly admit " We don't know all the math but we do
know the end-result ! "
Good engineering sense is defined as the
ability to quickly identify a problem, come up with practical solutions and then select the
most efficient option. Many real-world problems are independent of any complex mathematics,
but the same systematic scientific approach is required to solve them. The application of
sound logic is all that is required to reduce complexities to simplicities.
Early in the introduction I stated that it was an over-statement to claim that one needs
calculus to become a scientist. Those engineers and scientists who got by with a cursory
knowledge of the subject fall into the realm of the pseudo-scientist and charlatan. As
Ortega wrote, the pseudo-scientist, "advances the science which he scarcely knows, and
increases perforce the encyclopedia of knowledge of which he is conscientiously ignorant...".
It is the thrill and enlightenment of understanding nature that drives the scientist. The
engineer takes science a step further to control nature to suit man's needs. Both the
engineer and scientist share a deep appreciation for the workings of nature; an appreciation
that develops into wisdom.
Therefore, the purpose of studying calculus is two-fold. First is to introduce you to the
basic concepts of mathematics used to study almost any type of changing phenomena within a
controlled setting. Second, studying calculus will develop invaluable scientific sense and
practical engineering problem solving skills in you. You will understand how to think
logically to reduce even the most complex systems to a few interacting components. As you
study the main concepts, theories and examples in this book, your mind will develop into a
powerful systematic instrument.
To conclude on a humorous note, read the following satirical essay
from The Onion in the chapter on
1925 in their book, Our Dumb
Should we teach our children facts? No, I say, a thousand times no! As
they grow into tomorrow's farmers, housewives, mill workers, and coal
miners, facts are the last things they will need. Manners!
Subservience! Above all, obedience! To speak only when spoken to, and
not to cause trouble! These are the principles upon which our
educational system was grounded. Why in the name of God should we
replace it with a system that actually encourages an ignorant man to ask
questions? A good citizen does nothing of the sort. He is content with
the reasons he is given by his betters.
- Has God created man or man created God? Some say man and nature
are too well made to happen at random. The many physical laws of this
universe must be made by a creator just as traffic laws are made by man.
While evolution may appear random, it is actually an accumulation of
infinite changes over time. What we see is just the current state, not
the infinite layers that have been set down over eternity. For this
reason, it is easy to conclude the current state is too perfect to
understand. But when you study how each layer of evolution happens, it
becomes quite easy to understand how complexity can be achieved from the
simplest of changes accumulated over time.
Comment on this question with regards to your views on religion and science.
- The paradox of life is that one must suffer to enjoy it. Or consider the
quote by the father of the modern Olympic movement, "
The most important thing in the Olympic Games is not to win but to take
part, just as the most important thing in life is not the triumph but
the struggle. The essential thing is not to have conquered but to have
fought well. " What do you think? Is life about achievements or the journey to those
The world contains
thousands of cultures. Each culture has its own set of morals and
standards which the society tries to live by. What one
culture considers good, another culture may consider it taboo.
While most people
are fundamentally good, there is little
consistency among socially accepted behavior.
Are morals absolute or do they exist only relative to each
other? In other words do you think certain actions can be judged as pure good or
evil, regardless of personal upbringing?
What aspects of your culture
could be perceived as strange or negative to someone of a different culture?
Next section ->
Section 1.2 - Why Not to Study Calculus