Masani / Patel / Patil Elementary Calculus

1. Auflage 2014
ISBN: 978-1-4832-7489-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 352 Seiten, Web PDF

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.

Autoren/Hrsg.

Masani, P. R.

Patel, R. C.

Patil, D. J.

Boas, Ralph P.

Inhaltsverzeichnis

1;Front Cover;1
2;Elementary Calculus;4
3;Copyright Page;5
4;Foreword;6
5;Preface;8
6;List of Abbreviations and Symbols;11
7;Table of Contents;12
8;Chapter I. Numbers;18
8.1;1. Basic Laws of Numbers;18
8.2;2. The Integers and the Rational Numbers;20
8.3;3. Deficiencies of the Rational Number System;22
8.4;5. The Ordering of the Real Numbers; Absolute Value;26
8.5;6. Approximation of Irrational Numbers by Rational Numbers;29
9;Chapter II. Functions;31
9.1;1. Relation and Function;31
9.2;2. Sequences;36
9.3;3. Classification of Real Functions;37
9.4;4. The Graphs of Relations;42
9.5;5. Quantities;47
9.6;†6. Variables and Constants;49
10;Chapter III. Basic Ideas and Problems of the Calculus;51
10.1;1. Introduction;51
10.2;2. The Concept of Instantaneous Velocity;51
10.3;3. The Concept of Tangent;53
10.4;4. The Concept of Area;54
10.5;5. Basic ideas of the Calculus;56
11;Chapter IV. Limits and Continuity;58
11.1;1. Limits of Sequences;58
11.2;2. Limits of Functions Defined on Intervals;63
11.3;3. Theorems on Limits;69
11.4;4. Continuity;73
11.5;5. Plane Curves;79
11.6;6. The Numbers e and p;81
11.7;EXERCISES IV;84
12;Chapter V. Derivatives;87
12.1;1. Definition of the Derivative;87
12.2;2. Geometric Meaning of f'(x);90
12.3;3. Rates. The d/dx and Dot Notations;92
12.4;4. Rules of Differentiation;96
12.5;5. Derivatives of Polynomials and Rational Functions;100
12.6;6. Derivatives of the Trigonometric Functions;101
12.7;7. Derivatives of Higher Order;104
12.8;8. Primitives;106
12.9;EXERCISES V;107
13;Chapter VI. Differentiation of Composite, Inverse,and Implicitly Defined Functions;111
13.1;1. Composite Functions and Their Differentiation;111
13.2;2. The Inverse of a Function and Its Differentiation;116
13.3;3. The Inverse Trigonometric Relations;121
13.4;4. Functions Given Implicitly;126
13.5;5. Differentiation of Implicitly Defined Functions;129
13.6;EXERCISES VI;133
14;Chapter VIl. Geometrical Applications of Derivatives;135
14.1;1. Tangent to a Graph;135
14.2;2. Mean Value Theorem;140
14.3;3. Significance of the Signs of f' and f';140
14.4;4. Maxima, Minima, and Inflections;145
14.5;5. Graph Tracing;150
14.6;EXERCISES VII;155
15;Chapter VIII. Physical and Other Applications of Derivatives;158
15.1;1. Approximate Evaluations;158
15.2;2. Rates of Change;161
15.3;3. Coefficients of Elasticity and Diffusion;172
15.4;4. Problems In Maxima and Minima;175
15.5;EXERCISES VIII A;180
16;Chapter IX. Integration;186
16.1;1. Plan of This Chapter;186
16.2;2. The Definition of Area of an Ordinate Set;186
16.3;3. The Definite Integral; Physical Illustrations;191
16.4;4. Properties of the Definite Integral;198
16.5;5. The Indefinite Integral and Its Relation to the Primitive;201
17;Chapter .. Logarithmic, Exponential, and Power Functions;206
17.1;1. Review of the Theory of Exponents;206
17.2;2. The Standard Logarithmic Function;208
17.3;3. The Standard Exponential Function;212
17.4;4. General Exponential and Logarithmic Functions;214
17.5;5. The Power Function;218
17.6;6. Applications of the Exponential Function;219
17.7;EXERCISES X;222
18;Chapter XI. Primitives;225
18.1;1. Basic Principles;225
18.2;2. Integration by Parts;230
18.3;3. Integration by Substitution;235
18.4;4. Primitives of Rational Functions. Method of Partial Fractions;242
18.5;5. Miscellaneous Methods. Reduction Formulas;249
18.6;EXERCISES XI A;255
19;Chapter XII. Geometrical Applications of Integrals;258
19.1;1. Areas of Plane Regions;258
19.2;2. Volumes of Solids;267
19.3;3. Lengths of Plane Curves;275
19.4;4. Area of a Surface of Revolution;279
19.5;EXERCISES XII;285
20;Chapter XIII. Simple Differential Equations;288
20.1;1. What Are Differential Equations?;288
20.2;2. First Order Differential Equations;290
20.3;3. Second Order Differential Equations;297
20.4;4. Simple Harmonic Motion;304
20.5;5. Motion of a Projectile;307
21;APPENDIX I. Relations as Sets of Ordered Couples;310
22;APPENDIX II. Proofs of Theorems on Limits (IV, §3) and, of the Mean Value Theorem (VII, §2);314
23;APPENDIX III. Infinite Series;319
24;APPENDIX IV. Partial Derivatives;323
24.1;1. Rectangular Coordinate System in Space;323
24.2;2. The Graph of a Function of Two Variables;324
24.3;3. Partial Derivatives;324
25;APPENDIX V. Approximate Integration;330
25.1;1. Trapezoidal Rule;330
25.2;2. Simpson's Rule;331
25.3;3. Lemma;332
26;APPENDIX VI. Duhamers Principle;335
27;APPENDIX VII. The S-Notation;336
28;Books for Further Reading and Reference;338
29;ANSWERS;339
30;Index;350