| 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
|
| 2. SOME APPLICATIONS OF INTEGRATION
|
| 2.1 |
Introduction |
88
|
| 2.2 |
The area of a region between two graphs expressed as an integral |
88
|
| 2.3 |
Worked examples |
89
|
| 2.4 |
Exercises |
94
|
| 2.5 |
The trigonometric functions |
94
|
| 2.6 |
Integration formulas for the sine and cosine |
94
|
| 2.7 |
A geometric description of the sine and cosine functions |
94
|
| 2.8 |
Exercises |
94
|
| 2.9 |
Polar coordinates |
94
|
| 2.10 |
The integral for area in polar coordinates |
94
|
| 2.11 |
Exercises |
94
|
| 2.12 |
Application of integration to the calculation of volume |
94
|
| 2.13 |
Exercises |
94
|
| 2.14 |
Application of integration to the calculation of work |
94
|
| 2.15 |
Exercises |
94
|
| 2.16 |
Average value of a function |
94
|
| 2.17 |
Exercises |
94
|
| 2.18 |
The integral as a function of the upper limit. Indefinite integrals |
94
|
| 2.19 |
Exercises |
94
|
| 3. CONTINUOUS FUNCTIONS
|
| 3.1 |
Informal description of continuity |
126
|
| 3.2 |
The definition of the limit of a function |
127
|
| 3.3 |
The definition of continuity of a function |
130
|
| 3.4 |
The basic limit theorems. More examples of continuous functions |
131
|
| 3.5 |
Proofs of the basic limit theorems |
135
|
| 3.6 |
Exercises |
138
|
| 3.7 |
Composite functions and continuity |
140
|
| 3.8 |
Exercises |
142
|
| 3.9 |
Bolzano's theorem for continuous functions |
142
|
| 3.10 |
The intermediate-value theorem for continuous functions |
144
|
| 3.11 |
Exercises |
145
|
| 3.12 |
The process of inversion |
146
|
| 3.13 |
Properties of functions preserved by inversion |
147
|
| 3.14 |
Inverses of piecewise monotonic functions |
148
|
| 3.15 |
Exercises |
149
|
| 3.16 |
The extreme-value theorem for continuous functions |
150
|
| 3.17 |
The small-span theorem for continuous functions (uniform continuity) |
152
|
| 3.18 |
The integrability theorem for continuous functions |
152
|
| 3.19 |
Mean-value theorems for integrals of continuous functions |
154
|
| 3.20 |
Exercises |
155
|
| 4. DIFFERENTIAL CALCULUS
|
| 4.1 |
Historical introduction |
156
|
| 4.2 |
A problem involving velocity |
157
|
| 4.3 |
The derivative of a function |
159
|
| 4.4 |
Examples of derivatives |
161
|
| 4.5 |
The algebra of derivatives |
164
|
| 4.6 |
Exercises |
167
|
| 4.7 |
Geometric interpretation of the derivative as a slope |
169
|
| 4.8 |
Other notations for derivatives |
171
|
| 4.9 |
Exercises |
173
|
| 4.10 |
The chain rule for differentiating composite functions |
174
|
| 4.11 |
Applications of the chain rule. Related rates and implicit differentiation |
176
|
| 4.12 |
Exercises |
179
|
| 4.13 |
Applications of the differentiation to extreme values of cuntions |
181
|
| 4.14 |
The mean-value theorem for derivatives |
183
|
| 4.15 |
Exercises |
186
|
| 4.16 |
Applications of the mean-value theorem to geometric properties of functions |
187
|
| 4.17 |
Second-derivative test for extrema |
188
|
| 4.18 |
Curve sketching |
189
|
| 4.19 |
Exercises |
191
|
| 4.20 |
Worked examples of extremum problems |
191
|
| 4.21 |
Exercises |
194
|
| 4.22 |
Partial derivatives |
196
|
| 4.23 |
Exercises |
201
|
| 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION
|
| 5.1 |
The derivative of an indefinite integral. The first fundamental theorem of calculus |
202
|
| 5.2 |
The zero-derivative theorem |
204
|
| 5.3 |
Primitive functions and the second fundamental theorem of calculus |
205
|
| 5.4 |
Properties of a function deduced from properties of its derivative |
207
|
| 5.5 |
Exercises |
208
|
| 5.6 |
The Leibniz notation for primitives |
210
|
| 5.7 |
Integration by substitution |
212
|
| 5.8 |
Exercises |
216
|
| 5.9 |
Integration by parts |
217
|
| 5.10 |
Exercises |
220
|
| 5.11 |
Miscellaneous review exercises |
222
|
| 6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS
|
| 6.1 |
Introduction |
226
|
| 6.2 |
Motivation for the definition of the natural logarithm as an integral |
227
|
| 6.3 |
The definition of the logarithm. Basic properties |
226
|
| 6.4 |
The graph of the natural logarithm |
226
|
| 6.5 |
Consequences of the functional equation L(ab) = L(a) + L(b) |
226
|
| 6.6 |
Logarithms referred to any positive base \(b \ne 1\) |
226
|
| 6.7 |
Introduction |
226
|
| 6.8 |
Introduction |
226
|
| 6.9 |
Introduction |
226
|
| 6.10 |
Introduction |
226
|
| 6.11 |
Introduction |
226
|
| 6.12 |
Introduction |
226
|
| 6.13 |
Introduction |
226
|
| 6.14 |
Introduction |
226
|
| 6.15 |
Introduction |
226
|
| 6.16 |
Introduction |
226
|
| 6.17 |
Introduction |
226
|
| 6.18 |
Introduction |
226
|
| 6.19 |
Introduction |
226
|
| 6.20 |
Introduction |
226
|
| 6.21 |
Introduction |
226
|
| 6.22 |
Introduction |
226
|
| 6.23 |
Introduction |
226
|
| 6.24 |
Introduction |
226
|
| 6.25 |
Exercises |
267
|
| 6.26 |
Miscellaneous review exercises |
268
|
| 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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