Understanding Calculus

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  Testimonials
  Table of Contents

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

  Inverse Functions
  Exponents
  Exponential
  Functions
  Applications of
  Exponential
  Functions
  Sine and Cosine
  Function
  Sine Function
  Sine Function -
  Differentiation and
  Integration
  Oscillatory Motion
  Mean Value
  Theorem
  Taylor Series
  More Taylor Series
  Integration
  Techniques

  Links
  Contact

 
 
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Preface

The purpose of this book is to present mathematics as the science of deductive reasoning and not as the art of manipulation. Unfortunately, many students feel mathematics is incomprehensible and is riddled with complex and abstract jargon. My goal is to impose a lasting understanding of and appreciation for calculus on the student.

Unfortunately, students are rarely given any example of practical applications. The curriculum's idea of exercises is nothing more than sheer number-crunching and manipulation of variables. The entire underlying principle of order and beauty upon which calculus is based, is neglected. The problems never call upon the student's ability to think logically. Rather, they require no more than time and persistence, to allow the trial and error method to succeed.

It is clear that the above scenario leaves the student with no more than two options. One, they can burden their mind out in trying to make sense of this mysterious science, where theorems appear out of nowhere and seem to have no connection with previously learned material. Or second, they can choose to memorize. Without a doubt, the majority of students will opt for the latter. Not only is it less time-consuming, but it also yields the correct answer!

One may argue that it is not important for students to understand what they are learning. It is especially disheartening to see everyone around you not care about understanding knowledge. One should just learn to accept and tolerate it , rather than criticize and feel bad about it. Such is the way of the world. If you want to create a system you need to have the understanding, but to run a system ( which most people do ), just cook book procedures are all you need to know.

The over-emphasis on the calculator and foremostly the computer is yet another point of confusion for the student. The computer is only a time-saving machine whose usefulness depends on the knowledge of the user. I do admit the computer is a remarkable machine, yet it is this fascination that gives students a false sense of what they are doing. The confidence gained from all the correct answers leads to an inseparable dependence where the student is absolutely helpless without it.

It is all these shortcomings I set out to correct in writing my book. My book is intended to give the student an understanding of what calculus is truly about. Only by explaining where something has come from will I be able to show where it is going. It does not take more intelligence than that of a parrot to be able to go through a list of theorems and equations; but only when one understands their origins can one correctly and confidently apply them in the real world.

I assume absolutely nothing and neither do I take anything for granted. Each chapter has a definite beginning, followed by logical, elegant and clear proofs, concluded with a brief summary that ties everything together. My only reference has been reason and if you could find just three lines that remotely resemble the lines from any other book, I would feel greatly distressed.

Throughout the textbook I constantly refer to science and engineering. The purpose of this to show how the scientific method applies to all disciplines and to understand that mathematics is an expression of ones observations and hypothesis. I do not overdose the student with series of real-world applications where calculus is applied. Doing so would only succeed in showing that calculus and real-world applications are linked together by chance alone. My goal is to present mathematics through science. Therefore an emphasis is placed on mastering the scientific method of analysis through understanding the necessary concepts of differential and integral calculus.

My book begins with a brief yet important review of the number system, laws of arithmetic, and some algebra. I have used simple pictures to show the student these fundamental concepts and operations. My definition of a number culminates with a philosophical look at the difference between actions and objects and then explains how a number can represent either attribute. There are no rigorous definition muddled with Greek language and abstract symbols.

The goal is for the student to visually understand the operations he or she takes for granted though uses so extensively. It is not obvious why multiplication and division dominate most fundamental equations from engineering to biology. For example many students can not explain conceptually the difference between 2+2 and 2 X 2, much less why a fraction times another fraction results in smaller number. This is due to the students inability to relate the concept of the number with multiplication. The chapter I have written reduces all the operations of arithmetic with the philosophical definition I have given to the number as an independent entity.

After this chapter I begin my in-depth study of calculus with an introduction to the function, giving full and complete definitions that center on perceiving it as a mathematical relationship. I introduce the concept of dimension to show how a function defines a situation in terms of its interacting conditions or dimensions.

I then show how it is possible to graph a function in a two-dimensional system using the idea of the independent and dependent variable. From plotting a few points to plotting more points I intuitively show the behavior of a function is only as accurate as the numbers of points it has been evaluated at. Once again it is not obvious how a function can be graphed as continuous changing line. I then give a short proof, using the idea of a small change in x producing a small change in f(x) to explain how the graph of the function is the continuous line drawn through these points. This is all done with graphical examples followed by a complete explanation of what exactly is going on.

Having completed the chapter on functions and their graphs, I begin a thorough immersion into the analysis of the function and its graph. Beginning with a few pages just on the definition of rate of change, I move on to a long study of the graph of a function that is not constant. Of course I do not begin with the assumption that the slope is not constant. I divide the graph over an interval into small intervals Dx, over which the graph will be analyzed independently. Again using the idea of slopes I introduce the terms average and approximate rates of change. It is here where I explain the equation for the average rate of change through any two points on the graph:

Having established this equation I then proceed to analyze the graph in a more complete way by breaking it up into intervals of value D x equal to a small number. The goal is for the student to understand that we are now looking for a way of defining the instantaneous rate of change of the function. From there I develop the concept of taking the limit and showing exactly what is meant by tangent to a graph. Building upon the slope equation and my definition of the graph of a function as a line connecting a set of points I show what exactly is meant by having a unique line through one point. Once again I use clear and simple examples that contain a minimum of notation and focus on understanding the relation between instantaneous rate of change and calculus.

I then go on to derive the normal rules for differentiation, clearly explaining each step in the derivations. I take considerable time with the chain rule which derives itself from the definition of a function and the rules for differentiating them.

From there I move on to the graphical interpretation of the derivative and the second derivative. As always I go through a clear and complete explanations of concavity and by assuming nothing. The underlying goal of this chapter is for the student to understand the relationship between the function and its derivative. This is done by emphasizing that the behavior of a function can be entirely predicted just by studying its derivative, f(x).

Having developed this link between function and derivative I take a more numerical and thorough look at how the derivative alone can be used to accurately graph the original functions. First I show how the equation:

can be thought of as I take considerable time explaining the difference between the discrete and the differential . Using this new equation I explain linearization and errors. In the next section of this chapter I introduce the process of integration. By taking a graph and breaking it up into four continuous sub-intervals I show how the exact change in the function, f, over the entire interval can be found by evaluating the f(x) Dx over each sub-interval D x and then summing up the delta x f . In the discussion it becomes rather obvious how the exact d fs of the graph can only be found by letting our D x go to a smaller number and summing up the many f(x) D x over the interval. In explaining all this I only use two new symbols:

It is in my highly theoretical chapter on Integration do I delve into the meaning of this equation and introduce the integral sign between the two symbols. What I do in this chapter is basically undo all that I did with the derivative. Throughout my discussion of the derivative I show it as a subtracting and dividing process. In explaining integration I show it as a multiplying and adding process or the reverse of differentiation. The emphasis is placed on understanding the relation the derivative has to its integral in terms of the net change in a function f(x) from a to b can be found by evaluating the derivative times a small D x and summing up these values from a to b. I show intuitively how the error gets less and less as D x gets smaller by using areas. By presenting different simple proofs for the Fundamental theorem of calculus, there should be little doubt in the students mind as to what integration means and more importantly how it relates to concept of differential change.

The next chapter I go through a series of geometric applications from areas to arc length. As an Engineer I have seen that students are generally completely helpless when it comes to setting up integrals. This is entirely due to the fact that the student has no idea what it means to integrate a function although they are more than capable of integrating some of the most awkward looking functions. What this chapter does is draw the bridge between the theory in the previous chapter and the practical applications in the following chapter. Even though the student understands integration, applying it requires yet another theoretical chapter. The transition from f, x, F, M, P etc.. to ( the integral for finding the total angle of twist of a rod, length L, of varying radius with J as the polar moment of inertia subjected by a twisting torsion force, T) appear to have little to do with each other except that they are evaluated the same way.

In going through the geometric applications I leave behind all the xs, fs and f(x)'s and replace them with their geometric equivalents of length, l, width, w, and height, h. In order to understand how to set up integral I use the concept of a independent and dependent variable to explain when one constant is changing with respect to another constant a differential needs to be written. I do not delve too much into this theory of writing differentials as I leave it to the next chapter where identifying the differential is not a simple matter of . I concentrate on understanding the integral as a three step process of summation, integration and then evaluation. What is important is the method as the previous chapter already explained how these processes are all related.

It is in the final chapter on applications of integration that it becomes necessary to develop a consistent theory of integration that can be applied to any problem from engineering to economics. I further develop the idea of a constant being expressed as a function of another constant and then explaining how to identify the independent variable in the equation. I present a logical and reasonable algorithm for setting up integrals in problems.

This is perhaps the most important chapter in my book as its goal is to reconcile the theory of function, change, differentiation and integration all into one clear and concise method. This is really what calculus is all about. Modern textbooks devote most of their time to evaluating integrals while they completely disregard the numerous steps that must be taken before an integral can be set up and evaluated. A student must be able to understand how to set up an integral in a practical situation be it from electromagnetic forces to dynamic response of a skyscraper during an earthquake. Calculus is an utterly useless tool without this fundamental understanding of what integration is all about as the student will be able to play with calculus but he or she will never know how to use it.

This ends my text-book on calculus. My book is intended to offer an introduction to calculus for College or High-School. The course may be completed in either one semester or one year, depending how in depth the instructor is prepared to go. Throughout this Preface or introduction, I have constantly been referring to my complete discussions and explanations that are unlike the nonsense that accompanies theorems and concepts in other books. It is difficult to explain how exactly it is different for that would require my re-writing the book in these opening line. I can assure you that what awaits you is something unlike anything you have read and studied before.

More than 85% of the work is original and entirely conceived using the power of reason and logic. The remaining 15% is material that is covered in most every text. Though my presentation of such material has resemblance of form, in terms of content it is considerably different. Other books present a puzzle whereas I have taken the time to complete that puzzle not to mention adding further observations along the way to help strengthen the student's understanding.

It is therefore no overstatement to say that the entire book is unique and has almost no connection to the modern textbook. The only way to judge the validity of this statement is to continue on and read it. I am certain what you will learn will fascinate you.

Next section -> Chapter 1 - Why Study Calculus?

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