| Chapter/Section # |
Title |
Page #
|
| I. INTRODUCTION
|
| Part 1: Historical Introduction
|
| I 1.1 |
The two basic concepts of calculus |
1
|
| I 1.2 |
Historical background |
2
|
| I 1.3 |
The method of exhaustion for the area of a parabolic segment |
3
|
| *I 1.4 |
Exercises |
8
|
| I 1.5 |
A critical analysis of the Archimedes' method |
8
|
| I 1.6 |
The approach to calculus to be used in this book |
10
|
| Part 2: Some Basic Concepts of the Theory of Sets
|
| I 2.1 |
Introduction to set theory |
11
|
| I 2.2 |
Notations for designating sets |
12
|
| I 2.3 |
Subsets |
12
|
| I 2.4 |
Unions, intersections, complements |
13
|
| I 2.5 |
Exercises |
15
|
| Part 3: A set of Axioms for the Real-Number System
|
| I 3.1 |
Introduction |
17
|
| I 3.2 |
The field axioms |
17
|
| *I 3.3 |
Exercises |
19
|
| I 3.4 |
The order axioms |
19
|
| *I 3.5 |
Exercises |
21
|
| I 3.6 |
Integers and rational numbers |
21
|
| I 3.7 |
Geometric interpretation of real numbers as points on a line |
22
|
| I 3.8 |
Upper bound of a set, maximum element, least upper bound (supremum) |
23
|
| I 3.9 |
The least-Upper-bound axiom (completeness axiom) |
25
|
| I 3.10 |
The Archimedean property of the real-number system |
25
|
| I 3.11 |
Fundamental properties of the supremum and infimum |
26
|
| *I 3.12 |
Exercises |
28
|
| *I 3.13 |
Existence of square roots of nonnegative real numbers |
29
|
| *I 3.14 |
Roots of higher order. Rational powers |
30
|
| *I 3.15 |
Representation of real numbers by decimals |
30
|
| Part 4: Mathematical Induction, Summation Notation, and Related Topics
|
| I 4.1 |
An example of a proof by mathematical induction |
32
|
| I 4.2 |
The principle of mathematical induction |
34
|
| *I 4.3 |
The well-ordering principle |
34
|
| I 4.4 |
Exercises |
35
|
| *I 4.5 |
Proof of the well-ordering principle |
37
|
| I 4.6 |
The summation notation |
37
|
| I 4.7 |
Exercises |
39
|
| I 4.8 |
Absolute values and the triangle inequality |
41
|
| I 4.9 |
Exercises |
43
|
| *I 4.10 |
Miscellaneous exercises involving induction |
44
|
| 1. THE CONCEPTS OF INTEGRAL CALCULUS
|
| 1.1 |
The basic ideas of Cartesian geometry |
48
|
| 1.2 |
Functions. Informal description and examples |
50
|
| 1.3 |
Functions. Formal definition as a set of ordered pairs |
53
|
| 1.4 |
More examples of real functions |
54
|
| 1.5 |
Exercises |
56
|
| 1.6 |
The concept of area as a set function |
57
|
| 1.7 |
Exercises |
60
|
| 1.8 |
Intervals and ordinate sets |
60
|
| 1.9 |
Partitions and step functions |
61
|
| 1.10 |
Sum and product of step functions |
63
|
| 1.11 |
Exercises |
63
|
| 1.12 |
The definition of the integral for step functions |
64
|
| 1.13 |
Properties of the integral of a step function |
66
|
| 1.14 |
Other notations for integrals |
69
|
| 1.15 |
Exercises |
70
|
| 1.16 |
The integral of more general functions |
72
|
| 1.17 |
Upper and lower integrals |
74
|
| 1.18 |
The area of an ordinate set expressed as an integral |
75
|
| 1.19 |
Informal remarks on the theory and technique of integration |
75
|
| 1.20 |
Monotonic and piecewise monotonic functions. Definitions and examples |
76
|
| 1.21 |
Integrability of bounded monotonic functions |
77
|
| 1.22 |
Calculation of the integral of a bounded monotonic function |
79
|
| 1.23 |
Calculation of the integral \(\int_0^b x^p dx\) when \(p\) is a positive integer |
79
|
| 1.24 |
The basic properties of the integral |
80
|
| 1.25 |
Integration of polynomials |
81
|
| 1.26 |
Exercises |
83
|
| 1.27 |
Proofs of the basic properties of the integral |
84
|
| PART II: INTUITIVE GEOMETRY
|
| Chapter 5: Distance and Angles
|
| 1 |
Distance |
107
|
| 2 |
Angles |
110
|
| 3 |
The Pythagoras theorem |
120
|
| Chapter 6: Isometries
|
| 1 |
Some standard mappings of the plane |
133
|
| 2 |
Isometries |
143
|
| 3 |
Composition of isometries |
150
|
| 4 |
Inverse of isometries |
155
|
| 5 |
Characterization of isometries |
163
|
| 6 |
Congruences |
166
|
| Chapter 7: Area and Applications
|
| 1 |
Area of a disc of radius r |
173
|
| 2 |
Circumference of a circle of radius r |
180
|
| PART III: COORDINATE GEOMETRY
|
| Chapter 8: Coordinates and Geometry
|
| 1 |
Coordinate systems |
191
|
| 2 |
Distance between points |
197
|
| 3 |
Equation of a circle |
203
|
| 4 |
Rational points on a circle |
206
|
| Chapter 9: Operations on Points
|
| 1 |
Dilations and reflections |
213
|
| 2 |
Addition, subtraction, and the parallelogram law |
218
|
| Chapter 10: Segments, Rays, and Lines
|
| 1 |
Segments |
229
|
| 2 |
Rays |
231
|
| 3 |
Lines |
236
|
| 4 |
Ordinary equation for a line |
246
|
| Chapter 11: Trigonometry
|
| 1 |
Radian measure |
249
|
| 2 |
Sine and cosine |
252
|
| 3 |
The graphs |
264
|
| 4 |
The tangent |
266
|
| 5 |
Addition formulas |
272
|
| 6 |
Rotations |
277
|
| Chapter 12: Some Analytic Geometry
|
| 1 |
The straight line again |
281
|
| 2 |
The parabola |
291
|
| 3 |
The ellipse |
297
|
| 4 |
The hyperbola |
300
|
| 5 |
Rotation of hyperbolas |
305
|
| PART IV: MISCELLANEOUS
|
| Chapter 13: Functions
|
| 1 |
Definition of a function |
313
|
| 2 |
Polynomial functions |
318
|
| 3 |
Graphs of functions |
330
|
| 4 |
Exponential function |
333
|
| 5 |
Logarithms |
338
|
| Chapter 14: Mappings
|
| 1 |
Definition |
345
|
| 2 |
Formalism of mappings |
351
|
| 3 |
Permutations |
359
|
| Chapter 15: Complex Numbers
|
| 1 |
The complex plane |
375
|
| 2 |
Polar form |
380
|
| Chapter 16: Induction and Summations
|
| 1 |
Induction |
383
|
| 2 |
Summations |
388
|
| 3 |
Geometric series |
396
|
| Chapter 17: Determinants
|
| 1 |
Matrices |
401
|
| 2 |
Determinants of order 2 |
406
|
| 3 |
Properties of 2 x 2 determinants |
409
|
| 4 |
Determinants of order 3 |
414
|
| 5 |
Properties of 3 x 3 determinants |
418
|
| 6 |
Cramer's Rule |
424
|
| Index |
429
|
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