# Elementary Calculus

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Elementary Calculus presents a three semester introductory course on calculus. This book reveals the conceptual development of the calculus, taking into cognizance the technical and applied sides and standards of clarity and rigor that prevail in mathematics. The topics discussed include the basic laws of numbers, classification of real functions, and concept of instantaneous velocity. The limits of functions defined on intervals, derivatives of the trigonometric functions, and standard logarithmic function are also reviewed. This text likewise considers integration by substitution, lengths of plane curves, and simple harmonic motion. This publication is designed for students who have a knowledge of elementary trigonometry, and either have had a one semester course on analytic or coordinate geometry or might take such a course with calculus.

## Table of Contents

Foreword

Preface

List of Abbreviations and Symbols

Chapter I Numbers

1. Basic Laws of Numbers

2. The Integers and the Rational Numbers

3. Deficiencies of the Rational Number System

4. Real Numbers

5. The Ordering of the Real Numbers;Absolute Value

6. Approximation of Irrational Numbers by Rational Numbers

Chapter II Functions

1. Relation and Function

2. Sequences

3. Classification of Real Functions

4. The Graphs of Relations

5. Quantities

6. Variables and Constants

Chapter III Basic Ideas and Problems of the Calculus

1. Introduction

2. The Concept of Instantaneous Velocity

3. The Concept of Tangent

4. The Concept of Area

5. Basic Ideas of the Calculus

Chapter IV Limits and Continuity

1. Limits of Sequences

2. Limits of Functions Defined on Intervals

3. Theorems on Limits

4. Continuity

5. Plane Curves

6. The Numbers e and π

Exercises IV

Chapter V Derivatives

1. Definition of the Derivative

2. Geometric Meaning of ƒ'(x)

3. Rates. The d/dx and Dot Notations

4. Rules of Differentiation

5. Derivatives of Polynomials and Rational Functions

6. Derivatives of the Trigonometric Functions

7. Derivatives of Higher Order

8. Primitives

Exercises V

Chapter VI Differentiation of Composite, Inverse, and Implicitly Defined Functions

1. Composite Functions and Their Differentiation

2. The Inverse of a Function and Its Differentiation

3. The Inverse Trigonometric Relations

4. Functions Given Implicitly

5. Differentiation of Implicitly Defined Functions

Exercises VI

Chapter VII Geometrical Applications of Derivatives

1. Tangent to a Graph

2. Mean Value Theorem

3. Significance of the Signs of ƒ' and ƒ"

4. Maxima, Minima, and Inflections

5. Graph Tracing

Exercises VII

Chapter VIII Physical and Other Applications of Derivatives

1. Approximate Evaluations

2. Rates of Change

3. Coefficients of Elasticity and Diffusion

4. Problems in Maxima and Minima

Exercises VIII

Chapter IX Integration

1. Plan of This Chapter

2. The Definition of Area of an Ordinate Set

3. The Definite Integral; Physical Illustrations

4. Properties of the Definite Integral

5. The Indefinite Integral and Its Relation to the Primitive

Chapter Χ Logarithmic, Exponential, and Power Functions

1. Review of the Theory of Exponents

2. The Standard Logarithmic Function

3. The Standard Exponential Function

4. General Exponential and Logarithmic Functions

5. The Power Function

6. Applications of the Exponential Function

Exercises X

Chapter XI Primitives

1. Basic Principles

2. Integration by Parts

3. Integration by Substitution

4. Primitives of Rational Functions. Method of Partial Fractions

5. Miscellaneous Methods. Reduction Formulas

Exercises XI

Chapter XII Geometrical Applications of Integrals

1. Areas of Plane Regions

2. Volumes of Solids

3. Lengths of Plane Curves

4. Area of a Surface of Revolution

Exercises XII

Chapter XIII Simple Differential Equations

1. What Are Differential Equations?

2. First Order Differential Equations

3. Second Order DiflFerential Equations

4. Simple Harmonic Motion

5. Motion of a Projectile

Appendix I Relations as Sets of Ordered Couples

Appendix II Proofs of Theorems on Limits (IV, §3) and of the Mean Value Theorem (VII, §2)

Appendix III Infinite Series

Appendix IV Partial Derivatives

1. Rectangular Coordinate System in Space

2. The Graph of a Function of Two Variables

3. Partial Derivatives

Appendix V Approximate Integration

1. Trapezoidal Rule

2. Simpson's Rule

3. Lemma

Appendix VI Duhamel's Principle

Appendix VII The Ʃ-Notation

Books for Further Reading and Reference

Answers to Odd Numbered Exercises

Index

## Product details

- No. of pages: 352
- Language: English
- Copyright: © Academic Press 1965
- Published: January 1, 1965
- Imprint: Academic Press
- eBook ISBN: 9781483274898