Calculus of One Variable (eBook)

Stanley I. Grossman (Autor)

eBook Download: PDF

2014 | 2. Auflage
906 Seiten
Elsevier Science (Verlag)
978-1-4832-6246-8 (ISBN)

Infos

Calculus of One Variable, Second Edition presents the essential topics in the study of the techniques and theorems of calculus. The book provides a comprehensive introduction to calculus. It contains examples, exercises, the history and development of calculus, and various applications. Some of the topics discussed in the text include the concept of limits, one-variable theory, the derivatives of all six trigonometric functions, exponential and logarithmic functions, and infinite series. This textbook is intended for use by college students.

Inhalt

Front Cover 1
Calculus of One Variable 4
Copyright Page 5
Table of Contents 6
Preface 12
To the Instructor 19
Chapter 1. Preliminaries 24
1.1 SETS OF REAL NUMBERS 24
1.2 ABSOLUTE VALUE AND INEQUALITIES 30
1.3 THE CARTESIAN PLANE 36
1.4 LINES 42
1.5 EQUATIONS OF A STRAIGHT LINE 50
1.6 FUNCTIONS 58
1.7 OPERATIONS WITH FUNCTIONS 67
1.8 SHIFTING THE GRAPHS OF FUNCTIONS (OPTIONAL) 71
REVIEW QUESTIONS FOR CHAPTER 1 76
Chapter 2. Limiits and Derivatives 78
2.1 INTRODUCTION TO THE DERIVATIVE 78
2.2 THE CALCULATION OF LIMITS 86
2.3 THE LIMIT THEOREMS 101
2.4 INFINITE LIMITS AND LIMITS AT INFINITY 105
2.5 TANGENT LINES AND DERIVATIVES 116
2.6 THE DERIVATIVE AS A RATE OF CHANGE 129
2.7 CONTINUITY 135
2.8 THE THEORY OF LIMITS (OPTIONAL) 146
REVIEW EXERCISES FOR CHAPTER 2 157
Chapter 3. More atout Derivatives 160
3.1 SOME DIFFERENTIATION FORMULAS 160
3.2 THE PRODUCT AND QUOTIENT RULES 167
3.3 THE DERIVATIVE OF COMPOSITE FUNCTIONS: THE CHAIN RULE 173
3.4 THE DERIVATIVE OF A POWER FUNCTION 179
3.5 THE DERIVATIVES OF THE TRIGONOMETRIC FUNCTIONS 184
3.6 IMPLICIT DIFFERENTIATION 193
3.7 HIGHER-ORDER DERIVATIVES 199
3.8 APPROXIMATION AND DIFFERENTIALS 203
REVIEW EXERCISES FOR CHAPTER THREE 213
Chapter 4. Applications of the Derivative 215
4.1 RELATED RATES OF CHANGE 215
4.2 THE MEAN VALUE THEOREM 221
4.3 ELEMENTARY CURVE SKETCHING I: INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST 230
4.4 ELEMENTARY CURVE SKETCHING II: CONCAVITY AND THE SECOND DERIVATIVE TEST 241
4.5 THE THEORY OF MAXIMA AND MINIMA 254
4.6 MAXIMA AND MINIMA: APPLICATIONS 259
4.7 SOME APPLICATIONS IN ECONOMICS (Optional) 273
4.8 NEWTON'S METHOD FOR SOLVING EQUATIONS 280
REVIEW EXERCISES FOR CHAPTER FOUR 287
Chapter 5. The Intearal 288
5.1 INTRODUCTION 288
5.2 ANTIDERIVATIVES 291
5.3 THE S NOTATION 300
5.4 APPROXIMATIONS TO AREA 303
5.5 THE DEFINITE INTEGRAL 314
5.6 THE FUNDAMENTAL THEOREM OF CALCULUS 332
5.7 INTEGRATION BY SUBSTITUTION 343
5.8 THE AREA BETWEEN TWO CURVES 349
5.9 WORK, POWER, AND ENERGY (OPTIONAL) 356
5.10 ADDITIONAL INTEGRATION THEORY (OPTIONAL) 366
REVIEW EXERCISES FOR CHAPTER FIVE 370
Chapter 6. Exponentials anè Logarithms 372
6.1 INVERSE FUNCTIONS 373
6.2 THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS I 381
6.3 THE DERIVATIVES AND INTEGRALS OF loga x and ax 390
6.4 THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS II 398
6.5 DIFFERENTIATION AND INTEGRATION OF MORE GENERAL EXPONENTIAL AND LOGARITHMIC FUNCTIONS 413
6.6 DIFFERENTIAL EQUATIONS OF EXPONENTIAL GROWTH AND DECAY 422
6.7 APPLICATIONS IN ECONOMICS (OPTIONAL) 432
6.8 A MODEL FOR EPIDEMICS (OPTIONAL) 441
PROBLEMS 6.8 447
REVIEW EXERCISES FOR CHAPTER SIX 447
Chapter 7. More on Trionometric Function and the Hyperbolic functions 449
7.1 INTEGRATION OF TRIGONOMETRIC FUNCTIONS 449
7.2 THE INVERSE TRIGONOMETRIC FUNCTIONS 453
7.3 PERIODIC MOTION (OPTIONAL) 466
7.4 THE HYPERBOLIC FUNCTIONS 471
7.5 THE INVERSE HYPERBOLIC FUNCTIONS (OPTIONAL) 479
REVIEW EXERCISES FOR CHAPTER SEVEN 483
Chapter 8. Techniques of Integration 485
8.1 REVIEW OF THE BASIC FORMULAS OF INTEGRATION 486
8.2 INTEGRATION BY PARTS 487
8.3 INTEGRALS OF CERTAIN TRIGONOMETRIC FUNCTIONS 495
8.4 THE IDEA BEHIND INTEGRATION BY SUBSTITUTION 501
8.5 INTEGRALS INVOLVING va2 – x2, va2 + x2, AND vx2 -a2: TRIGONOMETRIC SUBSTITUTIONS 505
8.6 THE INTEGRATION OF RATIONAL FUNCTIONS I: LINEAR AND QUADRATIC DENOMINATORS 511
8.7 THE INTEGRATION OF RATIONAL FUNCTIONS II: THE METHOD OF PARTIAL FRACTIONS 519
8.8 OTHER SUBSTITUTIONS 527
8.9 USING THE INTEGRAL TABLES 532
8.10 NUMERICAL INTEGRATION 535
REVIEW EXERCISES FOR CHAPTER EIGHT 547
Chapter 9. Further Applications of the Defnite Intearal 550
9.1 VOLUMES 550
9.2 ARC LENGTH 565
9.3 SURFACE AREA 568
9.4 CENTER OF MASS AND THE FIRST MOMENT 573
9.5 THE CENTROID OF A PLANE REGION 576
9.6 MOMENTS OF INERTIA AND KINETIC ENERGY (OPTIONAL) 585
9.7 FLUID PRESSURE (OPTIONAL) 592
REVIEW EXERCISES FOR CHAPTER NINE 597
Chapter 10. Topics in Analytic Geometry 600
10.1 THE ELLIPSE AND TRANSLATION OF AXES 600
10.2 THE PARABOLA 609
10.3 THE HYPERBOLA 615
10.4 SECOND-DEGREE EQUATIONS AND ROTATION OF AXES 622
REVIEW EXERCISES FOR CHAPTER TEN 629
Chapter 11. Polar Coordinates 631
11.1 THE POLAR COORDINATE SYSTEM 631
11.2 GRAPHING IN POLAR COORDINATES 636
11.3 POINTS OF INTERSECTION OF GRAPHS OF POLAR EQUATIONS 646
11.4 DERIVATIVES AND TANGENT LINES 651
11.5 AREAS IN POLAR COORDINATES 657
REVIEW EXERCISES FOR CHAPTER ELEVEN 662
Chapter 12. Indeterminate forms and Improper Integrals 664
12.1 THE INDETERMINATE FORM 0/0 AND L'HÔPITAL'S RULE 664
12.2 PROOF OF L'HÔPITAL'S RULE (OPTIONAL) 669
12.3 OTHER INDETERMINATE FORMS 671
12.4 IMPROPER INTEGRALS 676
REVIEW EXERCISES FOR CHAPTER TWELVE 688
Chapter 13. Taylor Polynomials and Approximation 690
13.1 TAYLOR'S THEOREM AND TAYLOR POLYNOMIALS 690
13.2 A PROOF OF TAYLOR'S THEOREM, ESTIMATES ON THE REMAINDER TERM, AND A UNIQUENESS THEOREM (OPTIONAL) 697
13.3 APPROXIMATION USING TAYLOR POLYNOMIALS 703
REVIEW EXERCISES FOR CHAPTER THIRTEEN 711
Chapter 14. Sequences and Series 712
14.1 SEQUENCES OF REAL NUMBERS 712
14.2 BOUNDED AND MONOTONIC SEQUENCES 720
14.3 GEOMETRIC SERIES 725
14.4 INFINITE SERIES 731
14.5 SERIES WITH NONNEGATIVE TERMS I: TWO COMPARISON TESTS AND THE INTEGRAL TEST 737
14.6 SERIES WITH NONNEGATIVE TERMS II: THE RATIO AND ROOT TESTS 743
14.7 ABSOLUTE AND CONDITIONAL CONVERGENCE: ALTERNATING SERIES 747
14.8 POWER SERIES 755
14.9 DIFFERENTIATION AND INTEGRATION OF POWER SERIES 761
14.10 TAYLOR AND MACLAURIN SERIES 768
REVIEW EXERCISES FOR CHAPTER FOURTEEN 777
Appendixes 780
Appendix 1: Review oj Trigonometry 782
1.1 ANGLES AND RADIAN MEASURE 782
1.2 THE TRIGONOMETRIC FUNCTIONS AND BASIC IDENTITIES 786
1.3 OTHER TRIGONOMETRIC FUNCTIONS 791
1.4 TRIANGLES 794
Appendix 2: Mathematical Induction 798
Appendix 3: Determinants 803
Appendix 4: The Binomial Theorem 814
Appendix 5: The Proofs of Some Theorems on Limits, Continuity and Differentiation 818
TABLE 828
TABLE A.1 EXPONENTIAL FUNCTIONS 828
TABLE A.2 NATURAL LOGARITHMS 829
TABLE A.3 HYPERBOLIC FUNCTIONS 830
TABLE A.4 INTEGRALS 831
Answer to Odd-Numbered Problems and Review Exercises 841
Index 904

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 1-4832-6246-4 / 1483262464
• ISBN-13: 978-1-4832-6246-8 / 9781483262468