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Tables of Contents for Introductory Analysis
Chapter/Section Title
Page #
Page Count
Foundations of Real Analysis
Sets and Sentences
Sets
1
4
Open Sentences
5
5
Conditional Sentences
10
3
Real Numbers
Addition and Multiplication Properties of Real Numbers
13
6
Order and Inequalities
19
4
Absolute Value
23
5
Subsets of R
28
4
Review and Testing
Summary
32
1
Test
33
8
Analytic Geometry
Coordinate Geometry and Lines
The Cartesian Coordinate System: Distance and Midpoints
41
5
Lines
46
7
Conic Sections
Circles
53
3
Ellipses
56
7
Hyperbolas
63
6
Parabolas
69
4
Conic Sections
73
3
Intersections of Conic Section
76
4
Review and Testing
Summary
80
1
Test
81
8
Sequences, Series, and Limits
Mathematical Induction
Mathematical Induction
89
5
The Binomial Theorem
94
5
Sequences and Series
Recursive Definitions: Sequences and Series
99
4
Arithmetic Sequences
103
5
Geometric Sequences and Series
108
6
Limits and Sequences
Limits of Sequences
114
5
Infinite Series
119
6
The Axiom of Completeness
125
4
Integral and Rational Exponents
129
4
Review and Testing
Summary
133
2
Test
135
3
Cumulative Review
138
3
Functions and Limits
Functions
Functions and Functional Notation
141
5
The Algebra of Functions
146
5
Composition and Inversion of Functions
151
6
Limits of Functions
Limits and Their Properties
157
6
Continuous Functions
163
4
The Intermediate-Value Theorem
167
5
Limits Involving Infinity
172
5
Review and Testing
Summary
177
1
Test
178
7
Theory of Polynomial Equations
Complex Numbers
The Complex Field
185
4
The Complex Plane
189
4
Square Roots of Complex Numbers
193
3
General Theory of Polynomials
Polynomials Over Fields
196
5
The Fundamental Theorem of Algebra
201
4
Polynomials with Real Coefficients
Conjugate Imaginary Roots
205
3
Bounds For Real Roots; Descartes' Rule
208
3
Finding Rational Roots
211
4
Review and Testing
Summary
215
1
Test
216
5
Introduction to Differential Calculus
Differentiation
Derivatives of Functions
221
5
Properties of Derivatives
226
4
Notations for Derivatives
230
4
The Chain Rule
234
4
Applications
Using Derivatives in Graphing
238
4
Concavity and the Second Derivative
242
3
Applied Maxima and Minima
245
8
Derivatives as Rates of Change
253
5
Newton's Method
258
4
Review and Testing
Summary
262
1
Test
263
5
Cumulative Review
268
3
Trigonometric Functions and Triangle Solving
Circular Functions of Numbers
The Sine and Cosine Functions
271
6
The Other Circular Functions
277
4
Trigonometric Functions of Angles
Angles and Angle Measure
281
5
The Trigonometric Functions
286
4
Values of the Trigonometric Functions
290
2
Solving Triangles
Right Triangles
292
4
The Laws of Cosines and Sines
296
4
Solving General Triangles
300
5
Areas of Triangles
305
3
Review and Testing
Summary
308
1
Test
309
4
Trigonometric Identities and Graphs
Trigonometric Identities
Identities in One Variable
313
5
Trigonometric Addition Formulas
318
4
Double-Angle and Half-Angle Formulas
322
4
More Trigonometric Identities
326
4
Trigonometric Equations
330
4
Trigonometric Graphs
Periodicity and Other Properties
334
4
Sine and Cosine Graphs
338
5
More General Sine Curves
343
5
Other Trigonometric Graphs
348
2
Inverse Trigonometric Functions
350
5
Review and Testing
Summary
355
2
Test
357
4
Applications of Trigonometry
Applications Involving Polar Coordinates
The Polar Coordinate System
361
4
Graphs of Polar Equations
365
4
Polar Form of Complex Numbers
369
3
Powers and Roots of Complex Numbers
372
3
Parametric Equations
375
5
Calculus of Trigonometric Functions
Some Trigonometric Limits
380
4
Derivatives of the Trigonometric Functions
384
4
Rates and Extrema
388
5
Review and Testing
Summary
393
1
Test
394
5
Exponential and Logarithmic Functions
Exponential Functions
Real Exponents
399
4
Exponential Functions
403
5
The Natural Exponential Function
408
4
Logarithmic Functions
Logarithmic Functions
412
4
Laws of Logarithms
416
6
Derivatives
Derivatives of Exponential and Logarithmic Functions
422
5
Exponential Growth and Decay
427
5
Review and Testing
Summary
432
1
Test
433
5
Cumulative Review
438
3
Vectors
Vectors
Introduction to Vectors
441
5
Three-Dimensional Space
446
5
The Norm of a Vector
451
3
The Dot Product
454
3
Basis Vectors
457
5
Applications
Applications of Vectors
462
4
Vectors and Proofs
466
3
Vector Spaces
469
2
Review and Testing
Summary
471
1
Test
472
7
Matrices, Determinants, and Systems of Linear Equations
Transformations and Matrices
Linear Transformations
479
6
Matrices
485
5
Matrices of Linear Transformations
490
3
Rotation of Axes
493
4
Systems of Linear Equations
Linear Systems I
497
4
Linear Systems II
501
4
Determinants
505
5
The Inverse of a Matrix
510
5
Cramer's Rule
515
3
Review and Testing
Summary
518
1
Test
519
6
Further Vector Topics
Planes and Lines
The Cross Product
525
5
Planes
530
4
Lines
534
5
Surfaces
Quadric Surfaces
539
6
Cylindrical and Spherical Coordinates
545
4
Review and Testing
Summary
549
2
Test
551
3
Cumulative Review
554
3
Introduction to Integral Calculus
Antiderivatives and Definite Integrals
Antiderivatives
557
4
Integration by Substitution
561
4
Definite Integrals
565
3
Applications of Integration
Areas of Plane Regions
568
6
Volumes of Solids
574
6
Work and Energy
580
4
Review and Testing
Summary
584
1
Test
585
4
Probability and Statistics
Probability
Permutations
589
5
Combinations
594
5
Probability
599
4
Compound Events
603
7
Random Variables and Probability Distributions
610
6
Statistics
Statistical Measures
616
5
Variance and Standard Deviation
621
5
Normal Distributions
626
6
Confidence Intervals
632
3
Correlation
635
4
Review and Testing
Summary
639
2
Test
641
5
Using the Tables
646
1
Tables
647
22
Reference Materials
669
2
Glossary
671
9
Index
680
9
Credits
689
1
Appendix: The &epsis;-δ Definition of Limit
690