Introductory Analysis

by Dolciani, Mary P.; Sorgenfrey, Robert H.; Graham, John A.; Myers, David L.

• ISBN13: 9780395524329
• ISBN10: 0395524326
• Format: Hardcover
• Copyright: 1990-11-01
• Publisher: McDougal Littell/Houghton Mifflin

Table of Contents

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