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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Enables readers to apply the fundamentals of differential calculus to solve real-life problems in engineering and the physical sciences Introduction to Differential Calculus fully engages readers by presenting the fundamental theories and methods of differential calculus and then showcasing how the discussed concepts can be applied to real-world problems in engineering and the physical sciences. With its easy-to-follow style and accessible explanations, the book sets a solid foundation before advancing to specific calculus methods, demonstrating the connections between differential calculus theory and its applications. The first five chapters introduce underlying concepts such as algebra, geometry, coordinate geometry, and trigonometry. Subsequent chapters present a broad range of theories, methods, and applications in differential calculus, including: * Concepts of function, continuity, and derivative * Properties of exponential and logarithmic function * Inverse trigonometric functions and their properties * Derivatives of higher order * Methods to find maximum and minimum values of a function * Hyperbolic functions and their properties Readers are equipped with the necessary tools to quickly learn how to understand a broad range of current problems throughout the physical sciences and engineering that can only be solved with calculus. Examples throughout provide practical guidance, and practice problems and exercises allow for further development and fine-tuning of various calculus skills. Introduction to Differential Calculus is an excellent book for upper-undergraduate calculus courses and is also an ideal reference for students and professionals alike who would like to gain a further understanding of the use of calculus to solve problems in a simplified manner.
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Seitenzahl: 1069
Veröffentlichungsjahr: 2012
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Contents
Cover
Title Page
Copyright
Foreword
Preface
Biographies
Introduction
Acknowledgments
Chapter 1: From Arithmetic to Algebra
1.1 Introduction
1.2 The Set of Whole Numbers
1.3 The Set of Integers
1.4 The Set of Rational Numbers
1.5 The Set of Irrational Numbers
1.6 The Set of Real Numbers
1.7 Even and Odd Numbers
1.8 Factors
1.9 Prime and Composite Numbers
1.10 Coprime Numbers
1.11 Highest Common Factor (H.C.F.)
1.12 Least Common Multiple (L.C.M.)
1.13 The Language of Algebra
1.14 Algebra as a Language for Thinking
1.15 Induction
1.16 An Important Result: The Number of Primes is Infinite
1.17 Algebra as the Shorthand of Mathematics
1.18 Notations in Algebra
1.19 Expressions and Identities in Algebra
1.20 Operations Involving Negative Numbers
1.21 Division by Zero
Chapter 2: The Concept of a Function
2.1 Introduction
2.2 Equality of Ordered Pairs
2.3 Relations and Functions
2.4 Definition
2.5 Domain, Codomain, Image, and Range of a Function
2.6 Distinction Between “f ” and “f(x)”
2.7 Dependent and Independent Variables
2.8 Functions at a Glance
2.9 Modes of Expressing a Function
2.10 Types of Functions
2.11 Inverse Function f−1
2.12 Comparing Sets without Counting their Elements
2.13 The Cardinal Number of a Set
2.14 Equivalent Sets (Definition)
2.15 Finite Set (Definition)
2.16 Infinite Set (Definition)
2.17 Countable and Uncountable Sets
2.18 Cardinality of Countable and Uncountable Sets
2.19 Second Definition of an Infinite Set
2.20 The Notion of Infinity
2.21 An Important Note About the Size of Infinity
2.22 Algebra of Infinity (∞)
Chapter 3: Discovery of Real Numbers: Through Traditional Algebra
3.1 Introduction
3.2 Prime and Composite Numbers
3.3 The Set of Rational Numbers
3.4 The Set of Irrational Numbers
3.5 The Set of Real Numbers
3.6 Definition of a Real Number
3.7 Geometrical Picture of Real Numbers
3.8 Algebraic Properties of Real Numbers
3.9 Inequalities (Order Properties in Real Numbers)
3.10 Intervals
3.11 Properties of Absolute Values
3.12 Neighborhood of a Point
3.13 Property of Denseness
3.14 Completeness Property of Real Numbers
3.15 (Modified) Definition II (l.u.b.)
3.16 (Modified) Definition II (g.l.b.)
Chapter 4: From Geometry to Coordinate Geometry
4.1 Introduction
4.2 Coordinate Geometry (or Analytic Geometry)
4.3 The Distance Formula
4.4 Section Formula
4.5 The Angle of Inclination of a Line
4.6 Solution(s) of an Equation and its Graph
4.7 Equations of a Line
4.8 Parallel Lines
4.9 Relation Between the Slopes of (Nonvertical) Lines that are Perpendicular to One Another
4.10 Angle Between Two Lines
4.11 Polar Coordinate System
Chapter 5: Trigonometry and Trigonometric Functions
5.1 Introduction
5.2 (Directed) Angles
5.3 Ranges of sin θ and cos θ
5.4 Useful Concepts and Definitions
5.5 Two Important Properties of Trigonometric Functions
5.6 Graphs of Trigonometric Functions
5.7 Trigonometric Identities and Trigonometric Equations
5.8 Revision of Certain Ideas in Trigonometry
Chapter 6: More About Functions
6.1 Introduction
6.2 Function as a Machine
6.3 Domain and Range
6.4 Dependent and Independent Variables
6.5 Two Special Functions
6.6 Combining Functions
6.7 Raising a Function to a Power
6.8 Composition of Functions
6.9 Equality of Functions
6.10 Important observations
6.11 Even and Odd Functions
6.12 Increasing and Decreasing Functions
6.13 Elementary and Nonelementary Functions
Chapter 7a: The Concept of Limit of a Function
7a.1 Introduction
7a.2 Useful Notations
7a.3 The Concept of Limit of A Function: Informal Discussion
7a.4 Intuitive Meaning of Limit of A Function
7a.5 Testing The Definition [Applications of The ε, δ Definition of Limit]
7a.6 Theorem (B): Substitution Theorem
7a.7 Theorem (C): Squeeze Theorem Or Sandwich Theorem
7a.8 One-Sided Limits (Extension to the Concept of Limit)
Chapter 7b: Methods for Computing Limits of Algebraic Functions
7b.1 Introduction
7b.2 Methods for Evaluating Limits of Various Algebraic Functions
7b.3 Limit At Infinity
7b.4 Infinite Limits
7b.5 Asymptotes
Chapter 8: The Concept of Continuity of a Function, and Points of Discontinuity
8.1 Introduction
8.2 Developing the Definition of Continuity “At a Point”
8.3 Classification of the Points of Discontinuity: Types of Discontinuities
8.4 Checking Continuity of Functions Involving Trigonometric, Exponential, and Logarithmic Functions
8.5 From One-Sided Limit to One-Sided Continuity and its Applications
8.6 Continuity on an Interval
8.7 Properties of Continuous Functions
Chapter 9: The Idea of a Derivative of a Function
9.1 Introduction
9.2 Definition of the Derivative as a Rate Function
9.3 Instantaneous Rate of Change of y [=f(x)] at x = x1 and the Slope of its Graph at x = x1
9.4 A Notation for Increment(s)
9.5 The Problem of Instantaneous Velocity
9.6 Derivative of Simple Algebraic Functions
9.7 Derivatives of Trigonometric Functions
9.8 Derivatives of Exponential and Logarithmic Functions
9.9 Differentiability and Continuity
9.10 Physical Meaning of Derivatives
9.11 Some Interesting Observations
9.12 Historical Notes
Chapter 10: Algebra of Derivatives: Rules for Computing Derivatives of Various Combinations of Differentiable Functions
10.1 Introduction
10.2 Recalling the Operator of Differentiation
10.3 THE Derivative of a Composite Function
10.4 Usefulness of Trigonometric Identities in Computing Derivatives
10.5 Derivatives of Inverse Functions
Chapter 11a: Basic Trigonometric Limits and Their Applications in Computing Derivatives of Trigonometric Functions
11a.1 Introduction
11a.2 Basic Trigonometric Limits
11a.3 Derivatives of Trigonometric Functions
Chapter 11b: Methods of Computing Limits of Trigonometric Functions
11b.1 Introduction
11b.2 Limits of Type (I)
11b.3 Limits of the Type (II) [f(x), where a ≠ 0]
11b.4 Limits of Exponential and Logarithmic Functions
Chapter 12: Exponential Form(s) of a Positive Real Number and its Logarithm(s): Pre-Requisite for Understanding Exponential and Logarithmic Functions
12.1 Introduction
12.2 Concept of Logarithm
12.3 The Laws of Exponent
12.4 Laws of Exponents (or Laws of Indices)
12.5 Two Important Bases: “10” and “e”
12.6 Definition: Logarithm
12.7 Advantages of Common Logarithms
12.8 Change of Base
12.9 Why Were Logarithms Invented?
12.10 Finding A Common Logarithm of A (Positive) Number
12.11 Antilogarithm
12.12 Method of Calculation USING LOGARITHM
Chapter 13a: Exponential and Logarithmic Functions and Their Derivatives
13a.1 Introduction
13a.2 Origin of e
13a.3 Distinction Between Exponential and Power Functions
13a.4 The Value of e
13a.5 The Exponential Series
13a.6 Properties of e and Those of Related Functions
13a.7 Comparison of Properties of Logarithm(s) to the Bases 10 and e
13a.8 A Little More About e
13a.9 Graphs of Exponential Function(s)
13a.10 General Logarithmic Function
13a.11 Derivatives of Exponential and Logarithmic Functions
13a.12 Exponential Rate of Growth
13a.13 Higher Exponential Rates of Growth
13a.14 An Important Standard Limit
13a.15 Applications of the Function ex: Exponential Growth and Decay
Chapter 13b: Methods for Computing Limits of Exponential and Logarithmic Functions
13b.1 Introduction
13b.2 Review of Logarithms
13b.3 Some Basic Limits
13b.4 Evaluation of Limits Based on the Standard Limit
Chapter 14: Inverse Trigonometric Functions and Their Derivatives
14.1 Introduction
14.2 Trigonometric Functions (With Restricted Domains) and Their Inverses
14.3 The Inverse Cosine Function
14.4 The Inverse Tangent Function
14.5 Definition of the Inverse Cotangent Function
14.6 Formula for the Derivative of Inverse Secant Function
14.7 Formula for the Derivative of Inverse Cosecant Function
14.8 Important Sets of Results and their Applications
14.9 Application of Trigonometric Identities in Simplification of Functions and Evaluation of Derivatives of Functions Involving Inverse Trigonometric Functions
Chapter 15a: Implicit Functions and Their Differentiation
15a.1 Introduction
15a.2 Closer Look AT the Difficulties Involved
15a.3 The Method of Logarithmic Differentiation
15a.4 Procedure of Logarithmic Differentiation
Chapter 15b: Parametric Functions and Their Differentiation
15b.1 Introduction
15b.2 The Derivative of a Function Represented Parametrically
15b.3 Line of Approach for Computing the Speed of a Moving Particle
15b.4 Meaning of dy/dx with Reference to the Cartesian Form y = f(x) and Parametric Forms x = f(t), y = g(t) of the Function
15b.5 Derivative of One Function with Respect to the Other
Chapter 16: Differentials “dy” and “dx”: Meanings and Applications
16.1 Introduction
16.2 Applying Differentials to Approximate Calculations
16.3 Differentials of Basic Elementary Functions
16.4 Two Interpretations of the Notation dy/dx
16.5 Integrals in Differential Notation
16.6 To Compute (Approximate) Small Changes and Small Errors Caused in Various Situations
Chapter 17: Derivatives and Differentials of Higher Order
17.1 Introduction
17.2 Derivatives of Higher Orders: Implicit Functions
17.3 Derivatives of Higher Orders: Parametric Functions
17.4 Derivatives of Higher Orders: Product of Two Functions (Leibniz Formula)
17.5 Differentials of Higher Orders
17.6 Rate of Change of a Function and Related Rates
Chapter 18: Applications of Derivatives in Studying Motion in a Straight Line
18.1 Introduction
18.2 Motion in a Straight Line
18.3 Angular Velocity
18.4 Applications of Differentiation in Geometry
18.5 Slope of a Curve in Polar Coordinates
Chapter 19a: Increasing and Decreasing Functions and the Sign of the First Derivative
19a.1 Introduction
19a.2 The First Derivative Test for Rise and Fall
19a.3 Intervals of Increase and Decrease (Intervals of Monotonicity)
19a.4 Horizontal Tangents with a Local Maximum/Minimum
19a.5 Concavity, Points of Inflection, and the Sign of the Second Derivative
Chapter 19b: Maximum and Minimum Values of a Function
19b.1 Introduction
19b.2 Relative Extreme Values of a Function
19b.3 Theorem A
19b.4 Theorem B: Sufficient Conditions for the Existence of a Relative Extrema—In Terms of the First Derivative
19b.5 Sufficient Condition for Relative Extremum (In Terms of the Second Derivative)
19b.6 Maximum and Minimum of a Function on the Whole Interval (Absolute Maximum and Absolute Minimum Values)
19b.7 Applications of Maxima and Minima Techniques in Solving Certain Problems Involving the Determination of the Greatest and the Least Values
Chapter 20: Rolle's Theorem and the Mean Value Theorem (MVT)
20.1 Introduction
20.2 Rolle's Theorem (A Theorem on the Roots of a Derivative)
20.3 Introduction to the Mean Value Theorem
20.4 Some Applications of the Mean Value Theorem
Chapter 21: The Generalized Mean Value Theorem (Cauchy's MVT), L' Hospital's Rule, and their Applications
21.1 Introduction
21.2 Generalized Mean Value Theorem (Cauchy's MVT)
21.3 Indeterminate Forms and L'Hospital's Rule
21.4 L'Hospital's Rule (First Form)
21.5 L'Hospital's Theorem (For Evaluating Limits(s) of the Indeterminate Form 0/0.)
21.6 Evaluating Indeterminate Form of the Type ∞/∞
21.7 Most General Statement of L'Hospital's Theorem
21.8 Meaning of Indeterminate Forms
21.9 Finding Limits Involving Various Indeterminate Forms (by Expressing them in the Form 0/0 or ∞/∞)
Chapter 22: Extending the Mean Value Theorem to Taylor's Formula: Taylor Polynomials for Certain Functions
22.1 Introduction
22.2 The Mean Value Theorem For Second Derivatives: The First Extended MVT
22.3 Taylor's Theorem
22.4 Polynomial Approximations and Taylor's Formula
22.5 From Maclaurin Series To Taylor Series
22.6 Taylor's Formula for Polynomials
22.7 Taylor's Formula for Arbitrary Functions
Chapter 23: Hyperbolic Functions and Their Properties
23.1 Introduction
23.2 Relation Between Exponential and Trigonometric Functions
23.3 Similarities and Differences in the Behavior of Hyperbolic and Circular Functions
23.4 Derivatives of Hyperbolic Functions
23.5 Curves of Hyperbolic Functions
23.6 The Indefinite Integral Formulas for Hyperbolic Functions
23.7 Inverse Hyperbolic Functions
23.8 Justification for Calling sinh and cosh as Hyperbolic Functions Just as sine and cosine are Called Trigonometric Circular Functions
Appendix A: (Related To Chapter-2) Elementary Set Theory
A.1 Introduction
A.2 Elements of a Set
A.3 Set Notations
A.4 Specifying Sets
A.5 Singleton Set (or Unit Set)
A.6 The Null Set or the Empty Set
A.7 The Cardinal Number of a Set
A.8 Subset of a Set
A.9 Equality of Sets
A.10 Proper Subset
A.11 Comparability of Sets
A.12 Set of Sets
A.13 Power Set
A.14 Universal Set U
A.15 Operations on Sets
A.16 The Union (Logical Sum) of Two Sets A and B
A.17 The Intersection (Logical Product) of Two Sets A and B
A.18 Disjoint Sets
A.19 Difference of Two Sets A and B
A.20 Complement of a Set
Appendix B: (Related To Chapter-4)
B.1 Introduction
B.2
B.3 The Idea of a Double Napped Right Circular Cone and Conics
B.4 Conic Section: Definitions
B.5 Conics
B.6
B.7 Translation of Axes (or Shift of Origin)
Appendix C: (Related To Chapter-20)
Index
Copyright © 2012 by John Wiley & Sons. All rights reserved
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Library of Congress Cataloging-in-Publication Data:
Introduction to differential calculus: systematic studies with engineering applications for beginners / Ulrich L. Rohde... [et al.]. – 1st ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-118-11775-0 (hardback)
1. Differential calculus–Textbooks. I. Rohde, Ulrich L.
QA304.I59 2012
513′.33–dc232011018421
Foreword
“What is Calculus?” is a classic deep question. Calculus is the most powerful branch of mathematics, which revolves around calculations involving varying quantities. It provides a system of rules to calculate quantities, which cannot be calculated by applying any other branch of mathematics. Schools or colleges find it difficult to motivate students to learn this subject, while those who do take the course find it very mechanical. Many a times, it has been observed that students incorrectly solve real-life problems by applying Calculus. They may not be capable to understand or admit their shortcomings in terms of basic understanding of fundamental concepts! The study of Calculus is one of the most powerful intellectual achievements of the human brain. One important goal of this manuscript is to give beginner-level students an appreciation of the beauty of Calculus. Whether taught in a traditional lecture format or in the lab with individual or group learning, Calculus needs focusing on numerical and graphical experimentation. This means that the ideas and techniques have to be presented clearly and accurately in an articulated manner.
The ideas related with the development of Calculus appear throughout mathematical history, spanning over more than 2000 years. However, the credit of its invention goes to the mathematicians of the seventeenth century (in particular, to Newton and Leibniz) and continues up to the nineteenth century, when French mathematician Augustin-Louis Cauchy (1789–1857) gave the definition of the limit, a concept which removed doubts about the soundness of Calculus, and made it free from all confusion. The history of controversy about Calculus is most illuminating as to the growth of mathematics. The soundness of Calculus was doubted by the greatest mathematicians of the eighteenth century, yet, it was not only applied freely but great developments like differential equations, differential geometry, and so on were achieved. Calculus, which is the outcome of an intellectual struggle for such a long period of time, has proved to be the most beautiful intellectual achievement of the human mind.
There are certain problems in mathematics, mechanics, physics, and many other branches of science, which cannot be solved by ordinary methods of geometry or algebra alone. To solve these problems, we have to use a new branch of mathematics, known as Calculus. It uses not only the ideas and methods from arithmetic, geometry, algebra, coordinate geometry, trigonometry, and so on, but also the notion of limit, which is a new idea which, lies at the foundation of Calculus. Using this notion as a tool, the derivative of a function (which is a variable quantity) is defined as the limit of a particular kind.
In general, Differential Calculus provides a method for calculating “the rate of change” of the value of the variable quantity. On the other hand, Integral Calculus provides methods for calculating the total effect of such changes, under the given conditions. The phrase rate of change mentioned above stands for the actual rate of change of a variable, and not its average rate of change. The phrase “rate of change” might look like a foreign language to beginners, but concepts like rate of change, stationary point, and root, and so on, have precise mathematical meaning, agreed-upon all over the world. Understanding such words helps a lot in understanding the mathematics they convey. At this stage, it must also be made clear that whereas algebra, geometry, and trigonometry are the tools which are used in the study of Calculus, they should not be confused with the subject of Calculus.
This manuscript is the result of joint efforts by Prof. Ulrich L. Rohde, Mr. G. C. Jain, Dr. Ajay K. Poddar, and myself. All of us are aware of the practical difficulties of the students face while learning Calculus. I am of the opinion that with the availability of these notes, students should be able to learn the subject easily and enjoy its beauty and power. In fact, for want of such simple and systematic work, most students are learning the subject as a set of rules and formulas, which is really unfortunate. I wish to discourage this trend.
Professor Ulrich L. Rohde, Faculty of Mechanical, Electrical, and Industrial Engineering (RF and Microwave Circuit Design & Techniques) Brandenburg University of Technology, Cottbus, Germany has optimized this book by expanding it, adding useful applications, and adapting it for today's needs. Parts of the mathematical approach from the Rohde, Poddar, and Böeck textbook on wireless oscillators (The Design of Modern Microwave Oscillators for Wireless Applications: Theory and Optimization, John Wiley & Sons, ISBN 0-471-72342-8, 2005) were used as they combine differentiation and integration to calculate the damped and starting oscillation condition using simple differential equations. This is a good transition for more challenging tasks for scientific studies with engineering applications for beginners who find difficulties in understanding the problem-solving power of Calculus.
Mr. Jain is not an educator by profession, but his curiosity to go to the roots of the subject to prepare the so-called concept-oriented notes for systematic studies in Calculus is his contribution toward creating interest among students for learning mathematics in general, and Calculus in particular. This book started with these concept-oriented notes prepared for teaching students to face real-life engineering problems. Most of the material pertaining to this manuscript on calculus was prepared by Mr. G. C. Jain in the process of teaching his kids and helping other students who needed help in learning the subject. Later on, his friends (including me) realized the beauty of his compilation and we wanted to see his useful work published.
I am also aware that Mr. Jain got his notes examined from some professors at the Department of Mathematics, Pune University, India. I know Mr. Jain right from his scientific career at Armament Research and Development Establishment (ARDE) at Pashan, Pune, India, where I was a Senior Scientist (1982–1998) and headed the Aerodynamic Group ARDE, Pune in DRDO (Defense Research and Development Organization), India. Coincidently, Dr. Ajay K. Poddar, Chief Scientist at Synergy Microwave Corp., NJ 07504, USA was also a Senior Scientist (1990–2001) in a very responsible position in the Fuze Division of ARDE and was aware of the aptitude of Mr. Jain.
Dr. Ajay K. Poddar has been the main driving force towards the realization of the conceptualized notes prepared by Mr. Jain in manuscript form and his sincere efforts made timely publications possible. Dr. Poddar has made tireless effort by extending all possible help to ensure that Mr. Jain's notes are published for the benefit of the students. His contributions include (but are not limited to) valuable inputs and suggestions throughout the preparation of this manuscript for its improvement, as well as many relevant literature acquisitions. I am sure, as a leading scientist, Dr. Poddar will have realized how important it is for the younger generation to avoid shortcomings in terms of basic understanding of the fundamental concepts of Calculus.
I have had a long time association with Mr. Jain and Dr. Poddar at ARDE, Pune. My objective has been to proofread the manuscript and highlight its salient features. However, only a personal examination of the book will convey to the reader the broad scope of its coverage and its contribution in addressing the proper way of learning Calculus. I hope this book will prove to be very useful to the students of Junior Colleges and to those in higher classes (of science and engineering streams) who might need it to get rid of confusions, if any.
My special thanks goes to Dr. Poddar, who is not only a gifted scientist but has also been a mentor. It was his suggestion to publish the manuscript in two parts (Part I: Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners and Part II: Introduction to Integral Calculus: Systematic Studies with Engineering Applications for Beginners) so that beginners could digest the concepts of Differential and Integral Calculus without confusion and misunderstanding. It is the purpose of this book to provide a clear understanding of the concepts needed by beginners and engineers who are interested in the application of Calculus of their field of study. This book has been designed as a supplement to all current standard textbooks on Calculus and each chapter begins with a clear statement of pertinent definitions, principles, and theorems together with illustrative and other descriptive material. Considerably more material has been included here than can be covered in most high schools and undergraduate study courses. This has been done to make the book more flexible; to provide concept-oriented notes and stimulate interest in the relevant topics. I believe that students learn best when procedural techniques are laid out as clearly and simply as possible. Consistent with the reader's needs and for completeness, there are a large number of examples for self-practice.
The authors are to be commended for their efforts in this endeavor, and I am sure that both Part I and Part II will be an asset to the beginner's handbook on the bookshelf. I hope that after reading this book, the students will begin to share the enthusiasm of the authors in understanding and applying the principles of Calculus and its usefulness. With all these changes, the authors have not compromised our belief that the fundamental goal of Calculus is to help prepare beginners enter the world of mathematics, science, and engineering.
Finally, I would like to thank Susanne Steitz-Filler, Editor (Mathematics and Statistics) at John Wiley & Sons, Inc., Danielle Lacourciere, Senior Production Editor at John Wiley & Sons, Inc., and Sanchari Sil at Thomson Digital for her patience and splendid cooperation throughout the journey of this publication.
Ajoy Kanti GhoshProfessor & Faculty Incharge (Flight Laboratory)Department of Aerospace EngineeringIIT Kanpur, India
Preface
In general, there is a perception that Calculus is an extremely difficult subject, probably because the required number of good teachers and good books are not available. We know that books cannot replace teachers, but we are of the opinion that good books can definitely reduce dependence on teachers, and students can gain more confidence by learning most of the concepts on their own. In the process of helping students to learn Calculus, we have gone through many books on the subject and realized that whereas a large number of good books are available at the graduate level, there is hardly any book available for introducing the subject to beginners. The reason for such a situation can be easily understood by anyone who knows the subject of Calculus and hence the practical difficulties associated with the process of learning the subject. In the market hundreds of books are available on Calculus. All these books contain a large number of important solved problems. Besides, the rules for solving the problems and the list of necessary formulae are given in the books, without discussing anything about the basic concepts involved. Of course, such books are useful for passing the examination(s), but Calculus is hardly learnt from these books. Initially, the coauthors had compiled concept-oriented notes for systematic studies in differential and integral Calculus, intended for beginners. These notes were used by students in school- and undergraduate-level courses. The response and the appreciation experienced from the students and their parents encouraged us to make these notes available to the beginners. It is due to the efforts of our friends and well-wishers that our dream has now materialized in the form of two independent books: Part I for Differential Calculus and Part II for Integral Calculus. Of course there are some world class authors who have written useful books on the subject at introductory level, presuming that the reader has the necessary knowledge of prerequisites. Some such books are: What is calculus about? (By Professor WW Sawyer), Teach yourself calculus (By P. Abbott, B.A), Calculus Made Easy (By S.P. Thomson) and Calculus Explained (By W.J. Reichmann). Any person with some knowledge of Calculus will definitely appreciate the contents and the approach of the authors. However, a reader will be easily convinced that most of the beginners may not be able to get (from these books) the desired benefit, for various reasons. From this point of view, both Parts (Part-I & Part-II) of our book would prove to be unique since this provide comprehensive material on Calculus for the beginners. The first six chapters of Part-I would help the beginner to come up to the level, so that one can easily learn the concept of limit, which is in the foundation of calculus. The purpose of these works is to provide the basic (but solid) foundation of Calculus to beginners. The books aim to show them the enjoyment in the beauty and power of Calculus and develop the ability to select proper material needed for their studies in any technical and scientific field, involving Calculus.
One reason for such a high dropout rate is that at beginner levels, Calculus is so poorly taught. Classes tend to be so boring that students sometimes fall asleep. Calculus textbooks get fatter and fatter every year, with more multicolor overlays, computer graphics, and photographs of eminent mathematicians (starting with Newton and Leibniz), yet they never seem easier to comprehend. We look through them in vain for simple, clear exposition, and for problems that will hook a student's interest. Recent years have seen a great hue and cry in mathematical circle over ways to improve teaching Calculus to beginner and high-school students. Endless conferences have been held, many funded by the federal government, dozens of experimental programs are here and there. Some leaders of reform argue that a traditional textbook gets weightier but lacks the step-by-step approach to generate sufficient interest to learn Calculus in beginner, high school, and undergraduate students. Students see no reason why they should master tenuous ways of differentiating and integrating by hand when a calculator or computer will do the job. Leaders of Calculus reform are not suggesting that calculators and computers should no longer be used; what they observe is that without basic understanding about the subject, solving differentiation and integration problems will be a futile exercise. Although suggestions are plentiful for ways to improve Calculus understanding among students and professionals, a general consensus is yet to emerge.
The word “Calculus” is taken from Latin and it simply means a “stone” or “pebble,” which was employed by the Romans to assist the process of counting. By extending the meaning of the word “Calculus,” it is now applied to wider fields (of calculation) which involve processes other than mere counting. In the context of this book (with the discussion to follow), the word “Calculus” is an abbreviation for Infinitesimal Calculus or to one of its two separate but complimentary branches—Differential Calculus and Integral Calculus. It is natural that the above terminology may not convey anything useful to the beginner(s) until they are acquainted with the processes of differentiation and integration. What is the Calculus? What does it calculate? Is Calculus different from other branches of Mathematics? What type(s) of problems are handled by Calculus?
The author's aim throughout has been to provide a tour of Calculus for a beginner as well as strong fundamental basics to undergraduate students on the basis of the following questions, which frequently came to our minds, and for which we wanted satisfactory and correct answers.
i.What is Calculus?
ii.What does it calculate?
iii.Why do teachers of physics and mathematics frequently advise us to learn Calculus seriously?
iv.How is Calculus more important and more useful than algebra and trigonometry or any other branch of mathematics?
v.Why is Calculus more difficult to absorb than algebra or trigonometry?
vi.Are there any problems faced in our day-to-day life that can be solved more easily by Calculus than by arithmetic or algebra?
vii.Are there any problems which cannot be solved without Calculus?
viii.Why study Calculus at all?
ix.Is Calculus different from other branches of mathematics?
x.What type(s) of problems are handled by Calculus?
At this stage, we can answer these questions only partly. However, as we proceed, the associated discussions will make the answers clear and complete. To answer one or all of the above questions, it was necessary to know: How does the subject of Calculus begin?; How can we learn Calculus?, and What can Calculus do for us? The answers to these questions are hinted at in the books: What is Calculus about? and Mathematician's Delight, both by W.W. Sawyer. However, it will depend on the curiosity and the interest of the reader to study, understand, and absorb the subject. The author use very simple and nontechnical language to convey the ideas involved. However, if the reader is interested to learn the operations of Calculus faster, then he may feel disappointed. This is so, because the nature of Calculus and the methods of learning it are very different from those applicable in arithmetic or algebra. Besides, one must have a real interest to learn the subject, patience to read many books, and obtain proper guidance from teachers or the right books.
Calculus is the higher branch of mathematics, which enters into the process of calculating changing quantities (and certain properties), in the field of mathematics and various branches of science, including social science. It is said to be the Mathematics of Change. We cannot begin to answer any question related with change unless we know: What is that change and how it changes? This statement takes us closer to the concept of function y = f(x), wherein “y” is related to “x” through a rule “f.” We say that “y” is a function of x, by which we mean that “y” depends on “x.” (We say that “y” is a dependent variable, depending on the value of x, an independent variable.) From this statement it is clear that as the value of “x” changes, there results a corresponding change in the value of “y” depending on the nature of the function “f ” or the formula defining f.
The immense practical power of Calculus is due to its ability to describe and predict the behavior of the changing quantities “y” and “x.” In case of linear functions (which are of the form y = mx + b), an amount of change in the value of x causes a proportionate change in the value of y. However, in the cases of other functions (like y = x2 – 5, y = x3, y = x4 – x3 + 3, y = sin x, y = 3ex + x, etc.) which are not linear, no such proportionality exists. Our interest lies in studying the behavior of the dependent variable y[= f(x)] with respect to the change in (the value of) the independent variable “x.” In other words, we wish to find the rate at which “y” changes with respect to “x.”
We know that every rate is the ratio of change that may occur in the quantities, which are related to one another through a rule. It is easy to compute the average rate at which the value of y changes when x is changed from x1 to x2. It can be easily checked that (for the nonlinear functions) these average rate(s) are different between different values of x. [Thus, if |x2 – x1| = |x3 – x2| = |x4 – x3| = . . .. . ., (for all x1, x2, x3, x4, . . .) then we have f(x2) – f(x1) ≠ f(x3) – f(x2) ≠ f(x4) – f(x3) ≠ . . .. . .]. Thus, we get that the rate of change of y is different in between different values of x.
Our interest lies in computing the rate of change of “y” at every value of “x.” It is known as the instantaneous rate of change of “y” with respect to “x,” and we call it the “rate function” of “y” with respect to “x.” It is also called the derived function of “y” with respect to “x” and denoted by the symbol y′[=f ′(x)]. The derived function f ′(x) is also called the derivative of y[=f(x)] with respect to x. The equation y′ = f ′(x) tells that the derived function f′(x) is also a function of x, derived (or obtained) from the original function y = f(x). There is another (useful) symbol for the derived function, denoted by dy/dx. This symbol appears like a ratio, but it must be treated as a single unit, as we will learn later. The equation y′ = f ′(x) gives us the instantaneous rate of change of y with respect to x, for every value of “x,” for which f ′(x) is defined.
To define the derivative formally and to compute it symbolically is the subject of Differential Calculus. In the process of defining the derivative, various subtleties and puzzles will inevitably arise. Nevertheless, it will not be difficult to grasp the concept (of derivatives) with our systematic approach. The relationship between f(x) and f ′(x) is the main theme. We will study what it means for f ′(x) to be “the rate function” of f(x), and what each function says about the other. It is important to understand clearly the meaning of the instantaneous rate of change of f(x) with respect to x. These matters are systematically discussed in this book. Note that we have answered the first two questions and now proceed to answer the third one.
There are certain problems in mathematics and other branches of science, which cannot be solved by ordinary methods known to us in arithmetic, geometry, and algebra alone. In Calculus, we can study the properties of a function without drawing its graph. However, it is important to be aware of the underlying presence of the curve of the given function. Recall that this is due to the introduction of coordinate geometry by Decartes and Fermat. Now, consider the curve defined by the function y = x3 − x2 − x. We know that, the slope of this curve changes from point to point. If it is desired to find its slope at x = 2, then Calculus alone can help us give the answer, which is 7. No other branch of mathematics would be useful.
Calculus uses not only the ideas and methods from arithmetic, geometry, algebra, coordinate geometry, trigonometry, and so on, but also the notion of limit, which is a new idea that lies at the foundation of Calculus. Using the notion of limit as a tool, the derivative of a function is defined as the limit of a particular kind. (It will be seen later that the derivative of a function is generally a new function.) Thus, Calculus provides a system of rules for calculating changing quantities which cannot be calculated otherwise. Here it may be mentioned that the concept of limit is equally important and applicable in Integral Calculus, which will be clear when we study the concept of the definite integral in Chapter 5 of Part II. Calculus is the most beautiful and powerful achievement of the human brain. It has been developed over a period of more than 2000 years. The idea of derivative of a function is among the most important concepts in all of mathematics and it alone distinguishes Calculus from the other branches of mathematics.
Thederivative and an integral have found many diverse uses. The list is very long and can be seen in any book on the subject. Differential calculus is a subject which can be applied to anything that moves, or changes or has a shape. It is useful for the study of machinery of all kinds - for electric lighting and wireless, optics, and thermodynamics. It also helps us to answer questions about the greatest and smallest values a function can take. Professor W.W. Sawyer, in his famous book Mathematician's Delight, writes: Once the basic ideas of differential calculus have been grasped, a whole world of problems can be tackled without great difficulty. It is a subjectwell worth learning.
On the other hand, integral calculus considers the problem of determining a function from the information about its rate of change. Given a formula for the velocity of a body, as a function of time, we can use integral calculus to produce a formula that tells us how far the body has traveled from its starting point, at any instant. It provides methods for the calculation of quantities such as areas and volumes of curvilinear shapes. It is also useful for the measurement of dimensions of mathematical curves.
The concepts basic to Calculus can be traced, in uncrystallized form, to the time of the ancient Greeks (around 287–212 BC). However, it was only in the sixteenth and the early seventeenth centuries that mathematicians developed refined techniques for determining tangents to curves and areas of plane regions. These mathematicians and their ingenious techniques set the stage for Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716), who are usually credited with the “invention” of Calculus.
Later on the concept of the definite integral was also developed. Newton and Leibniz recognized the importance of the fact that finding derivatives and finding integrals (i.e., antiderivatives) are inverse processes, thus making possible the rule for evaluating definite integrals. All these matters are systematically introduced in Part II of the book. (There were many difficulties in the foundation of the subject of Calculus. Some problems reflecting conflicts and doubts on the soundness of the subject are reflected in “Historical Notes” given at the end of Chapter 9 of Part I.) During the last 150 years, Calculus has matured bit by bit. In the middle of the nineteenth century, French Mathematician Augustin-Louis Cauchy (1789–1857) gave the definition of limit, which removed all doubts about the soundness of Calculus and made it free from all confusion. It was then that, Calculus had become, mathematically, much as we know it today.
Around the year 1930, the increasing use of Calculus in engineering and sciences, created a necessary requirement to encourage students of engineering and science to learn Calculus. During those days, Calculus was considered an extremely difficult subject. Many authors came up with introductory books on Calculus, but most students could not enjoy the subject, because the basic concepts of the Calculus and its interrelations with the other subjects were probably not conveyed or understood properly. The result was that most of the students learnt Calculus only as a set of rules and formulas. Even today, many students (at the elementary level) learn Calculus in the same way. For them, it is easy to remember formulae and apply them without bothering to know: How the formulae have come and why do they work?
The best answer to the question “Why study Calculus at all?” is available in the book: Calculus from Graphical, Numerical and Symbolic Points of View by Arnold Ostebee and Paul Zorn. There are plenty of good practical and “educational” reasons, which emphasize that one must study Calculus.
Because it is good for applications;Because higher mathematics requires it;Because its good mental training;Because other majors require it; andBecause jobs require it.Also, another reason to study Calculus (according to the authors) is that Calculus is among our deepest, richest, farthest-reaching, and most beautiful intellectual achievements. This manuscript differs in certain respects, from the conventional books on Calculus for the beginners.
In both the Parts of the book (Part-I & Part-II), efforts have been made to ensure that the beginners do not face such situations. The concepts related with calculus and the interrelations between other subjects contributing towards learning calculus have been discussed in a simple language in both part of book (Part-I & Part-II), maintaining the interest and the enthusiasm of the reader. One such example is that of co-ordinate geometry, which is the merging of geometry with algebra and helps in visualizing an equation as representing a curve and vice-versa (Remember, calculus cannot be imagined without co-ordinate geometry.)
It is a fact that people can achieve many things in life even without learning calculus. It is really a big loss to all those who had an opportunity to learn calculus but unfortunately missed it for mere comfort and carelessness. Also, they would never know what really they have missed. It is hoped that this book will motivate the readers who may like to revise their basic knowledge of calculus to achieve the delayed benefit now.
Organization
The work is divided into two independent books: Book I—Differential Calculus (Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners) and Book II—Integral Calculus (Introduction to Integral Calculus: Systematic Studies with Engineering Applications for Beginners).
Part I consists of 23 chapters in which certain chapters are divided into two sub-units such as 7a and 7b, 11a and 11b, 13a and 13b, 15a and 15b, 19a and 19b. Basically, these sub-units are different from each other in one way, but they are interrelated through concepts. Also, there are Appendices A, B, and C for Part-I.
Part II consists of 9 chapters in which certain chapters are divided into two sub-units such as 3a and 3b, 4a and 4b, 6a and 6b, 7a and 7b, 8a and 8b, and finally 9a and 9b. The division of chapters is based on the same principle as in the case of Part I. Each chapter (or unit) in both the parts begins with an introduction, clear statements of pertinent definitions, principles, and theorems. Meaning(s) of different theorems and their consequences are discussed at length, before they are proved. The solved examples serve to illustrate and amplify the theory, thus bringing into sharp focus many fine points, to make the reader comfortable.
Illustrative and other descriptive material (along with notes and remarks) is given in each chapter to help the beginner understand the ideas involved. The CONTENTS of each chapter are reflected with all necessary details. Hence, it is not felt necessary to repeat the same details again. However, the following two points are worth emphasizing.
The Part-I (Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners):
The first six chapters of Part I are devoted for revising the prerequisites useful for both the parts. The selection of the material and its sequencing is very important. The reader will find it quite interesting and easy to absorb. Once the reader has gone through these chapters carefully, the reader will be fully prepared to study the concept of limit in Chapters 7a and 7b. The reader will not find any difficulty in absorbing and appreciating the ε – δ definition of limit. This definition is generally considered very difficult by the students and therefore it is mugged up without understanding its meaning.Chapter 8 deals with the concept of continuity that can be easily learnt, once the concept of limit is properly understood. (Chapters 7a, 7b, and 8 are considered as prerequisites for the purpose of understanding the concept of derivative.)Chapter 9 deals with the concept of derivative and its definition including the method of computing the derivative, by the first principle of a given function using the definition of derivative. (The concepts of limit, continuity, and derivative are discussed at length in the above chapters and must be studied carefully and with patience.) Once the reader has reached upto chapter-9, 50% ideas related with differential calculus is being understood. Subsequently, the ideas related with the integral calculus will be found very simple for understanding in Part-II of the book.Chapter 10 deals with the algebra of derivatives offering different methods for computing derivatives of functions depending on their properties and the algebra of limits. The concepts discussed in the remaining chapters do not pose problems to the reader since every concept is introduced in a proper sequence suggesting its necessity and applications.Chapter 11 is sub-divided into two part (11a and 11b). Chapter 11a deals with basic understanding of the trigonometric limits and its application for computing the derivatives of these functions.Chapter 11b deals with the methods of computing limits of trigonometric functions.Chapter 12 deals with exponential form (s) of a positive real number and its logarithm(s): Prerequisite for understanding exponential and logarithmic functions.Chapter 13 is sub-divided into two part (13a and 13b). Chapter 13a deals with the properties of exponential and logarithmic functions including their derivatives.Chapter 13b deals with methods for computing limits of exponential and logarithmic functionsChapter 14 deals with the inverse trigonometric functions and their properties including derivatives of many other functions using trigonometric identities.Chapter 15 is sub-divided into two part (15a and 15b).Chapter 15a deals with implicit functions and their differentiation.Chapter 15a deals with parametric functions and their differentiation.Chapter 16 deals with the concept of differentials dy and dx, and their applications in the process of integration and for understanding differential equations. It is also discussed how the symbol dy/dx for the derivative of a function can be looked upon as a ratio of differential dy to dx.Chapter 17 deals with the derivatives of higher order, their meaning and usefulness. Chapter 18 deals with applications of derivatives in studying motion in a straight line.Chapter 19 is sub-divided into two part (19a and 19b). Chapter 19a deals with the concepts of increasing and decreasing functions, studied using derivatives of first and second order.Chapter 19b deals with the methods of finding maximum and minimum values of a function using the concept of increasing and decreasing functions.Chapters 20, 21, and 22 are extremely important dealing with Mean Value Theorems and their applications like L'Hospital's Rule and introduction to the expansion of simple functions.Chapter 23 deals with the introduction of hyperbolic functions and their properties.Important advice for using both the parts of this book:
The CONTENTS clearly indicate how important it is to go through the prerequisites. Certain concepts [like (−1) · (−1) = 1, and why division by zero is not permitted in mathematics, etc] which are generally accepted as rules, are discussed logically. The concept of infinity and its algebra are very important for learning calculus. The ideas and definitions of functions introduced in Chapter-2, and extended in Chapter-6, are very useful.
The role of co-ordinate geometry in defining trigonometric functions and in the development of calculus should be carefully learnt.
The theorems, in both the Parts are proved in a very simple and convincing way. The solved examples will be found very useful by the students of plus-two standard and the first year college. Difficult problems have been purposely not included in solved examples and the exercise, to maintain the interest and enthusiasm of the beginners. The readers may pickup difficult problems from other books, once they have developed interest in the subject.
Concepts of limit, continuity and derivative are discussed at length in chapters 7(a) & 7(b), 8 and 9, respectively. The one who goes through from chapters-1 to 9 has practically learnt more than 60 % of differential calculus. The readers will find that remaining chapters of differential calculus are easy to understand. Subsequently, readers should not find any difficulties in learning the concepts of integral calculus and the process of integration including the methods of computing definite integrals and their applications in fining areas and volumes, etc.
The differential equations right from their formation and the methods of solving certain differential equations of first order and first degree will be easily learnt.
Students of High Schools and Junior College level may treat this book as a text book for the purpose of solving the problems and may study desired concepts from the book treating it as a reference book. Also the students of higher classes will find this book very useful for understanding the concepts and treating the book as a reference book for this purpose. Thus, the usefulness of this book is not limited to any particular standard. The reference books are included in the bibliography.
I hope, above discussion will be found very useful to all those who wish to learn the basics of calculus (or wish to revise them) for their higher studies in any technical field involving calculus.
Suggestions from the readers for typos/errors/improvements will be highly appreciated.
Finally, efforts have been made to ensure that interest of the beginner is maintained all through. It is fact that reading mathematics is very different from reading a novel. However, we hope that the readers will enjoy this book like a novel and learn Calculus. We are very sure that if beginners go through first six chapters of Part I (i.e., prerequisites), then they may not learn Calculus, but will start loving mathematics.
Dr. -Ing. Ajay Kumar PoddarChief Scientist,Synergy Microwave Corporation.NJ 07504, USA.Former Senior Scientist (Defense Research &Development Organization (DRDO), INDIA
Spring 2011
Biographies
Ulrich L. Rohde holds a Ph.D. in Electrical Engineering (1978) and a Sc.D. (Hon., 1979) in Radio Communications, a Dr.-Ing (2004), a Dr.-Ing Habil (2011), and several honorary doctorates. He is President of Communications Consulting Corporation; Chairman of Synergy Microwave Corp., Paterson, NJ; and a partner of Rohde & Schwarz, Munich, Germany. Previously, he was the President of Compact Software, Inc., and Business Area Director for Radio Systems of RCA, Government Systems Division, NJ. Dr. Rohde holds several dozen patents and has published more than 200 scientific papers in professional journals, has authored and coauthored 10 technical books. Dr. Rohde is a Fellow Member of the IEEE, Invited Panel Member for the FCC's Spectrum Policy Task Force on Issues Related to the Commission's Spectrum Policies, ETA KAPPA NU Honor Society, Executive Association of the Graduate School of Business-Columbia University, New York, the Armed Forces Communications & Electronics Association, fellow of the Radio Club of America, and former Chairman of the Electrical and Computer Engineering Advisory Board at New Jersey Institute of Technology. He is elected to the “First Microwave & RF Legends” (Global Voting from professionals and academician from universities and industries: Year 2006). Recently Prof. Rohde received the prestigious “Golden Badge of Honor” and university's highest Honorary Senator Award in Munich, Germany.
G.C. Jain graduated in science (Major—Advance Mathematics) from St. Aloysius College, Jabalpur in 1962. Mr. Jain has started his career as a Technical Supervisor (1963–1970), worked for more than 38 years as a Scientist in Defense Research & Development Organization (DRDO). He has been involved in many state-of-the-art scientific projects and also responsible for streamlining MMG group in ARDE, Pune. Apart from scientific activities, Mr. Jain spends most of his time as a volunteer educator to teach children from middle and high school.
Ajay K. Poddar graduated from IIT Delhi, Doctorate (Dr.-Ing.) from TU-Berlin (Technical University Berlin) Germany. Dr. Poddar is a Chief Scientist, responsible for design and development of state-of-the-art technology (oscillator, synthesizer, mixer, amplifier, filters, antenna, and MEMS based RF & MW components) at Synergy Microwave Corporation, NJ. Previously, he worked as a Senior Scientist and was involved in many state-of-the-art scientific projects in DRDO, India. Dr. Poddar holds more than dozen US, European, Japanese, Russian, Chinese patents, and has published more than 170 scientific papers in international conferences and professional journals, contributed as a coauthor of three technical books. He is a recipient of several scientific achievement awards, including RF & MW state-of-the-art product awards for the year 2004, 2006, 2008, 2009, and 2010. Dr. Poddar is a senior member of professional societies IEEE (USA), AMIE (India), and IE (India) and involved in technical and academic review committee, including the Academic Advisory Board member Don Bosco Institute of Technology, Bombay, India (2009–to date). Apart from academic and scientific activities, Dr. Poddar is involved in several voluntary service organizations for the greater cause and broader perspective of the society.
A.K. Ghosh graduated and doctorate from IIT Kanpur. Currently, he is a Professor & Faculty Incharge (Flight Laboratory) Accountable Manager (DGCA), Aerospace Engineering, IIT Kanpur, India (one of the most prestigious institutes in the world). Dr. Ghosh has published more than 120 scientific papers in international conferences and professional journals; recipient of DRDO Technology Award, 1993, young scientist award, Best Paper Award—In-house Journal “Shastra Shakti” ARDE, Pune. Dr. Ghosh has supervised more than 30 Ph.D. students and actively involved in several professional societies and board member of scientific review committee in India and abroad. Previously, he worked as a Senior Scientist and Headed Aerodynamic Group ARDE, Pune in DRDO, India.
Introduction
In less than 15 min, let us realize that calculus is capable of computing many quantities accurately, which cannot be calculated using any other branch of mathematics.
To be able to appreciate this fact, we consider a “nonvertical line” that makes an angle “θ” with the positive direction of x-axis, and that θ ≠ 0. We say that the given line is “inclined” at an angle “θ” (or that the inclination of the given line is “θ”).
The important idea of our interest is the “slope of the given line,” which is expressed by the trigonometric ratio “tan θ.” Technically the slope of the line tells us that if we travel by “one unit,” in the positive direction along the x-axis, then the number of units by which the height of the line rises (or falls) is the measure of its slope.
Also, it is important to remember that the “slope of a line” is a constant for that line. On the other hand “the slope of any curve” changes from point to point and it is defined in terms of the slope of the “tangent line” existing there. To find the slope of a curve y = f(x) at any value of x, the “differential calculus” is the only branch of Mathematics, which can be used even if we are unable to imagine the shape of the curve.
At this stage, it is very important to remember (in advance) and understand clearly that whereas, the subject of Calculus demands the knowledge of algebra, geometry, coordinate geometry and trigonometry, and so on (as a prerequisite), but they do not form the subject of Calculus. Hence, calculus should not be confused as a combination of these branches.
Calculus is a different subject. The backbone of Calculus is the “concept of limit,” which is introduced and discussed at length in Part I of the book. The first eight chapters in Part I simply offer the necessary material, under the head: What must you know to learn Calculus? We learn the concept of “derivative” in Chapter 9. In fact, it is the technical term for the “slope.”
The ideas developed in Part I are used to define an inverse operation of computing antiderivative. (In a sense, this operation is opposite to that of computing the derivative of a given function.)
Most of the developments in the field of various sciences and technologies are due to the ideas developed in computing derivatives and antiderivatives (also called integrals). The matters related with integrals are discussed in “Integral Calculus.”
The two branches are in fact complimentary, since the process of integral calculus is regarded as the inverse process of the differential calculus. As an application of integral calculus, the area under a curve y = f(x) from x = a to x = b, and the x-axis can be computed only by applying the integral calculus. No other branch of mathematics is helpful in computing such areas with curved boundaries.
Prof. Ulrich L. Rohde
Acknowledgments
There have been numerous contributions by many people to this work, which took much longer than expected. As always, Wiley has been a joy to work with through the leadership, patience, and understanding of Susanne Steitz-Filler, Jacqueline Palmieri, and Danielle Lacourciere. We would like to express our sincere appreciation to all the staff at John Wiley & Sons involved in this publishing project for their cheerful professionalism and outstanding supports.
It is a pleasure to acknowledge our indebtedness to Prof. Hemant Bhate (Department of Mathematics), Prof. M. S. Prasad, and Dr. Sukratu Barve (Center for Modeling and Simulation), University of Pune, India, who read the manuscript and gave valuable suggestions for improvements.
