Understanding Calculus

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  Table of Contents

  1. Why Study
  2. Numbers
  3. Functions
  4. The Derivative
  5. Differentiation
  6. Applications
  7. Free Falling
  8. Understanding
  9. Derivative
  10. Integration
  11. Understanding
  12. Differentials

  Inverse Functions
  Applications of
  Sine and Cosine
  Sine Function
  Sine Function -
  Differentiation and
  Oscillatory Motion
  Mean Value
  Taylor Series
  More Taylor Series


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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 not cure.

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 view.

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 interests and 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 world...".

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 ignorant..."

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:

  1. Completely ununderstandable
  2. Vague and indefinite
  3. Something correct that is obvious and self-evident, worked out by a long and difficult analysis and presented as an important discovery
  4. A claim based on the stupidity of the author that some obvious and correct thing accepted and checked for years is in fact false
  5. An attempt to do something probably impossible but of certainly no utility which it is finally revealed at the end fails
  6. 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 mathematics.

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 Century.

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.


  1. 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.

  2. 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 achievements?

  3. 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


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