| 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 |
229
|
| 6.4 |
The graph of the natural logarithm |
230
|
| 6.5 |
Consequences of the functional equation \(L(ab) = L(a) + L(b)\) |
230
|
| 6.6 |
Logarithms referred to any positive base \(b \ne 1\) |
232
|
| 6.7 |
Differentiation and integration formulas involving logarithms |
233
|
| 6.8 |
Logarithmic differentiation |
235
|
| 6.9 |
Exercises |
236
|
| 6.10 |
Polynomial approximations to the logarithm |
236
|
| 6.11 |
Exercises |
242
|
| 6.12 |
The exponential function |
242
|
| 6.13 |
Exponentials expressed as powers of e |
242
|
| 6.14 |
The definition of \(e^x\) for arbitrary real x |
244
|
| 6.15 |
The definition of \(a^x\) for \(a > 0\) and x real |
245
|
| 6.16 |
Differentiation and integration formulas involving exponentials |
245
|
| 6.17 |
Exercises |
248
|
| 6.18 |
The hyperbolic functions |
251
|
| 6.19 |
Exercises |
251
|
| 6.20 |
Derivatives of inverse functions |
252
|
| 6.21 |
Inverses of the trigonometric functions |
253
|
| 6.22 |
Exercises |
256
|
| 6.23 |
Integration by partial fractions |
258
|
| 6.24 |
Integrals which can be transformed into integrals of rational functions |
264
|
| 6.25 |
Exercises |
267
|
| 6.26 |
Miscellaneous review exercises |
268
|
| 7. POLYNOMIAL APPROXIMATIONS TO FUNCTIONS
|
| 7.1 |
Introduction |
272
|
| 7.2 |
The Taylor polynomials generated by a function |
273
|
| 7.3 |
Calculus of Taylor polynomials |
275
|
| 7.4 |
Exercises |
278
|
| 7.5 |
Taylor's formula with remainder |
278
|
| 7.6 |
Estimates for the error in Taylor's formula |
280
|
| 7.7 |
Other forms of the remainder in Taylor's formula |
283
|
| 7.8 |
Exercises |
284
|
| 7.9 |
Further remarks on the error in Taylor's formula. The o-notation |
286
|
| 7.10 |
Applications to indeterminate forms |
289
|
| 7.11 |
Exercises |
290
|
| 7.12 |
L'Hopital's rule for the indeterminate form 0/0 |
292
|
| 7.13 |
Exercises |
295
|
| 7.14 |
The symbols \(+\inf\) and \(-\inf\). Extension of L'Hopital's rule |
296
|
| 7.15 |
Infinite limits |
298
|
| 7.16 |
The behavior of log\(x\) and \(e^x\) for large \(x\) |
300
|
| 7.17 |
Exercises |
303
|
| 8. INTRODUCTION TO DIFFERENTIAL EQUATIONS
|
| 8.1 |
Introduction |
305
|
| 8.2 |
Terminology and notation |
306
|
| 8.3 |
A first-order differential equation for the exponential function |
307
|
| 8.4 |
First-order linear differential equations |
308
|
| 8.5 |
Exercises |
311
|
| 8.6 |
Some physical problems leading to first-order linear differential equations |
313
|
| 8.7 |
Exercises |
319
|
| 8.8 |
Linear equations of second order with constant coefficients |
322
|
| 8.9 |
Existence of solutions of the equation \(y^{} + by = 0\) |
323
|
| 8.10 |
Reduction of the general equation to the special case \(y^{} + by = 0\) |
324
|
| 8.11 |
Uniqueness theorem for the equation \(y^{} + by = 0\) |
324
|
| 8.12 |
Complete solution of the equation \(y^{} + by = 0\) |
326
|
| 8.13 |
Complete solution of the equation \(y^{} + ay^' + by = 0\) |
326
|
| 8.14 |
Exercises |
328
|
| 8.15 |
Nonhomogeneous linear equations of second order with constant coefficients |
329
|
| 8.16 |
Special methods for determining a particular solution of the nonhomogeneous equation \(y^{} + ay^' + by = R\) |
332
|
| 8.17 |
Exercises |
333
|
| 8.18 |
Examples of physical problems leading to linear second-order equations with constant coefficients |
334
|
| 8.19 |
Exercises |
339
|
| 8.20 |
Remarks concerning nonlinear differential equations |
339
|
| 8.21 |
Integral curves and direction fields |
341
|
| 8.22 |
Exercises |
344
|
| 8.23 |
First-order separable equations |
345
|
| 8.24 |
Exercises |
347
|
| 8.25 |
Homogeneous first-order equations |
347
|
| 8.26 |
Exercises |
350
|
| 8.27 |
Some geometrical and physical problems leading to first-order equations |
351
|
| 8.28 |
Miscellaneous review exercises |
355
|
| 9. COMPLEX NUMBERS
|
| 9.1 |
Historical introduction |
358
|
| 9.2 |
Definitions and field properties |
358
|
| 9.3 |
The complex numbers as an extension of the real numbers |
360
|
| 9.4 |
The imaginary unit \(i\) |
361
|
| 9.5 |
Geometric interpretation. Modulus and argument |
362
|
| 9.6 |
Exercises |
365
|
| 9.7 |
Complex exponentials |
366
|
| 9.8 |
Complex-valued functions |
368
|
| 9.9 |
Examples of differentiation and integration formulas |
369
|
| 9.10 |
Exercises |
371
|
| 10. SEQUENCES, INFINITE SERIES, IMPROPER INTEGRALS
|
| 10.1 |
Zeno's paradox |
374
|
| 10.2 |
Sequences |
378
|
| 10.3 |
Monotonic sequences of real numbers |
381
|
| 10.4 |
Exercises |
382
|
| 10.5 |
Infinite series |
383
|
| 10.6 |
The linearity property of convergent series |
385
|
| 10.7 |
Telescoping series |
386
|
| 10.8 |
The geometric series |
388
|
| 10.9 |
Exercises |
391
|
| 10.10 |
Exercises on decimal expansions |
393
|
| 10.11 |
Tests for convergence |
394
|
| 10.12 |
Comparison tests for series of nonnegative terms |
394
|
| 10.13 |
The integral test |
397
|
| 10.14 |
Exercises |
398
|
| 10.15 |
The root test and the ratio test for series of nonnegative terms |
399
|
| 10.16 |
Exercises |
402
|
| 10.17 |
Alternating series |
403
|
| 10.18 |
Conditional and absolute convergence |
406
|
| 10.19 |
The convergence tests of Dirichlet and Abel |
407
|
| 10.20 |
Exercises |
409
|
| 10.21 |
Rearrangements of series |
411
|
| 10.22 |
Miscellaneous review exercises |
414
|
| 10.23 |
Improper integrals |
416
|
| 10.24 |
Exercises |
420
|
| 11. SEQUENCES AND SERIES OF FUNCTIONS
|
| 11.1 |
Pointwise convergence of sequences of functions |
422
|
| 11.2 |
Uniform convergence of sequences of functions |
423
|
| 11.3 |
Uniform convergence and continuity |
424
|
| 11.4 |
Uniform convergence and integration |
425
|
| 11.5 |
A sufficient condition for uniform convergence |
427
|
| 11.6 |
Power series. Circle of convergence |
428
|
| 11.7 |
Exercises |
430
|
| 11.8 |
Properties of functions represented by real power series |
431
|
| 11.9 |
The Taylor's series generated by a function |
434
|
| 11.10 |
A sufficient condition for convergence of a Taylor's series |
435
|
| 11.11 |
Power-series expansions for the exponential and trigonometric functions |
435
|
| 11.12 |
Bernstein's theorem |
437
|
| 11.13 |
Exercises |
438
|
| 11.14 |
Power series and differential equations |
439
|
| 11.15 |
The binomial series |
441
|
| 11.16 |
Exercises |
443
|
| 12. VECTOR ALGEBRA
|
| 12.1 |
Historical introduction |
445
|
| 12.2 |
The vector space of n-tuples of real numbers |
446
|
| 12.3 |
Geometric interpretation for \(n \leq 3\) |
448
|
| 12.4 |
Exercises |
450
|
| 12.5 |
The dot product |
451
|
| 12.6 |
Length or norm of a vector |
453
|
| 12.7 |
Orthogonality of vectors |
455
|
| 12.8 |
Exercises |
456
|
| 12.9 |
Projections. Angle between vectors in n-space |
457
|
| 12.10 |
The unit coordinate vectors |
458
|
| 12.11 |
Exercises |
460
|
| 12.12 |
The linear span of a finite set of vectors |
462
|
| 12.13 |
Linear independence |
463
|
| 12.14 |
Bases |
466
|
| 12.15 |
Exercises |
467
|
| 12.16 |
The vector space \(V_N(C)\) of n-tuples of complex numbers |
468
|
| 12.17 |
Exercises |
470
|
| 13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY
|
| 1 |
Introduction |
471
|
| 2 |
Lines in n-space |
472
|
| 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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