Elementary Calculus

ForewordPrefaceList of Abbreviations and SymbolsChapter 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 NumbersChapter II Functions 1. Relation and Function 2. Sequences 3. Classification of Real Functions 4. The Graphs of Relations 5. Quantities 6. Variables and ConstantsChapter 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 CalculusChapter 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 p Exercises IVChapter V Derivatives 1. Definition of the Derivative 2. Geometric Meaning of f'(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 VChapter 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 VIChapter VII Geometrical Applications of Derivatives 1. Tangent to a Graph 2. Mean Value Theorem 3. Significance of the Signs of f' and f" 4. Maxima, Minima, and Inflections 5. Graph Tracing Exercises VIIChapter 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 VIIIChapter 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 PrimitiveChapter ¿ 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 XChapter 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 XIChapter 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 XIIChapter 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 ProjectileAppendix I Relations as Sets of Ordered CouplesAppendix II Proofs of Theorems on Limits (IV, §3) and of the Mean Value Theorem (VII, §2)Appendix III Infinite SeriesAppendix IV Partial Derivatives 1. Rectangular Coordinate System in Space 2. The Graph of a Function of Two Variables 3. Partial DerivativesAppendix V Approximate Integration 1. Trapezoidal Rule 2. Simpson's Rule 3. LemmaAppendix VI Duhamel's PrincipleAppendix VII The ¿-NotationBooks for Further Reading and ReferenceAnswers to Odd Numbered ExercisesIndex