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Index
Cover
Title Page
Copyright Page
Contents
Preface
I - Algebra and Analytic Geometry
1 Prologue
1.1 The Importance of Mathematics
1.2 The Uniqueness of Mathematics
1.3 The Unreasonable Effectiveness of Mathematics
1.4 Mathematics as a Language
1.5 What is Mathematics?
1.6 Mathematical Rigor
1.7 Advice to You
1.8 Remarks on Learning the Course
References
2 The Integers
2.1 The Integers
2.2 On Proving Theorems
2.3 Mathematical Induction
2.4 The Binomial Theorem
2.5 Mathematical Induction Using Undetermined Coefficients
2.6 The Ellipsis Method
2.7 Review and Fallacies in Algebra
2.8 Summary
3 Fractions—Rational Numbers
3.1 Rational Numbers
3.2 Euclid’s Algorithm
3.3 The Rational Number System
3.4 Irrational Numbers
3.5 On Finding Irrational Numbers
3.6 Decimal Representation of a Rational Number
3.7 Inequalities
3.8 Exponents—An Application of Rational Numbers
3.9 Summary and Further Remarks
4 Real Numbers, Functions, and Philosophy
4.1 The Real Line
4.2 Philosophy
4.3 The Idea of a Function
4.4 The Absolute Value Function
4.5 Assumptions About Continuity
4.6 Polynomials and Integers
4.7 Linear Independence
4.8 Complex Numbers
4.9 More Philosophy
4.10 Summary
5 Analytic Geometry
5.1 Cartesian Coordinates
5.2 The Pythagorean Distance
5.3 Curves
5.4 Linear Equations—Straight Lines
5.5 Slope
5.6 Special Forms of the Straight Line
5.7 On Proving Geometric Theorems in Analytic Geometry
5.8* The Normal Form of the Straight Line
5.9 Translation of the Coordinate Axes
5.10* The Area of a Triangle
5.11* A Problem in Computer Graphics
5.12 The Complex Plane
5.13 Summary
6 Curves of Second Degree—Conics
6.1 Strategy
6.2 Circles
6.3 Completing the Square
6.4 A More General Form of the Second-Degree Equation
6.5 Ellipses
6.6 Hyperbolas
6.7 Parabolas
6.8 Miscellaneous Cases
6.9* Rotation of the Coordinate Axes
6.10* The General Analysis
6.11 Symmetry
6.12 Nongeometric Graphing
6.13 Summary of Analytic Geometry
II - The Calculus of Algebraic Functions
7 Derivatives in Geometry
7.1 A History of the Calculus
7.2 The Idea of a Limit
7.3 Rules for Using Limits
7.4 Limits of Functions—Missing Values
7.5 The ∆ Process
7.6 Composite Functions
7.7 Sums of Powers of x
7.8 Products and Quotients
7.9 An Abstraction of Differentiation
7.10 On the Formal Differentiation of Functions
7.11 Summary
8 Geometric Applications
8.1 Tangent and Normal Lines
8.2 Higher Derivatives—Notation
8.3 Implicit Differentiation
8.4 Curvature
8.5 Maxima and Minima
8.6 Inflection Points
8.7 Curve Tracing
8.8 Functions, Equations, and Curves
8.9 Summary
9 Nongeometric Applications
9.1 Scaling Geometry
9.2 Equivalent Ideas
9.3 Velocity
9.4 Acceleration
9.5 Simple Rate Problems
9.6 More Rate Problems
9.7 Newton’s Method for Finding Zeros
9.8 Multiple Zeros
9.9 The Summation Notation
9.10 Generating Identities
9.11 Generating Functions—Place Holders
9.12 Differentials
9.13 Differentials Are Small
9.14 Summary
10 Functions of Several Variables
10.1 Functions of Two Variables
10.2 Quadratic Equations
10.3 Partial Derivatives
10.4 The Principle of Least Squares
10.5 Least-Squares Straight Lines
10.6 n-Dimensional Space
10.7 Test for Minima
10.8 General Case of Least-Squares Fitting
10.9 Summary
11 Integration
11.1 History
11.2 Area
11.3 The Area of a Circle
11.4 Areas of Parabolas
11.5 Areas in General
11.6 The Fundamental Theorem of the Calculus
11.7 The Mean Value Theorem
11.8 The Cauchy Mean Value Theorem
11.9 Some Applications of the Integral
11.10 Integration by Substitution
11.11 Numerical Integration
11.12 Summary
12 Discrete Probability
12.1 Introduction
12.2 Trials
12.3 Independent and Compound Events
12.4 Permutations
12.5 Combinations
12.6 Distributions
12.7 Maximum Likelihood
12.8 The Inclusion–Exclusion Principle
12.9 Conditional Probability
12.10 The Variance
12.11 Random Variables
12.12 Summary
13 Continuous Probability
13.1 Probability Density
13.2 A Monte Carlo Estimate of Pi
13.3 The Mean Value Theorem for Integrals
13.4 The Chebyshev Inequality
13.5 Sums of Independent Random Variables
13.6 The Weak Law of Large Numbers
13.7 Experimental Evidence for the Model
13.8 Examples of Continuous Probability Distributions
13.9 Bertrand’s Paradox
13.10 Summary
III - The Transcendental Functions and Applications
14 The Logarithm Function
14.1 Introduction—A New Function
14.2* In x Is Not an Algebraic Function
14.3 Properties of the Function ln x
14.4 An Alternative Derivation—Compound Interest
14.5 Formal Differentiation and Integration Involving ln x
14.6 Applications
14.7 Integration by Parts
14.8* The Distribution of Numbers
14.9 Improper Integrals
14.10 Systematic Integration
14.11 Summary
15 The Exponential Function
15.1 The Inverse Function
15.2 The Exponential Function
15.3 Some Applications of the Exponential Function
15.4 Stirling’s Approximation to n!
15.5 Indeterminate Forms
15.6 The Exponential Distribution
15.7 Random Events in Time
15.8 Poisson Distributions
15.9 The Normal Distribution
15.10 Normal Distribution, Maximum Likelihood, and Least Squares
15.11 The Gamma Function
15.12 Systematic Integration
15.13 Summary
16 The Trigonometric Functions
16.1 Review of the Trigonometric Functions
16.2 A Particular Limit
16.3 The Derivative of Sin x
16.4* An Alternative Derivation
16.5 Derivatives of the Other Trigonometric Functions
16.6 Integration Formulas
16.7 Some Definite Integrals of Importance
16.8 The Inverse Trigonometric Functions
16.9 Probability Problems
16.10 Summary of the Integration Formulas
16.11 Summary
17 Formal Integration
17.1 Purpose of This Chapter
17.2 Partial Fractions—Linear Factors
17.3 Quadratic Factors
17.4 Rational Functions in Sine and Cosine
17.5 Powers of Sines and Cosines
17.6 Integration by Parts—Reduction Formulas
17.7 Change of Variable
17.8 Quadratic Irrationalities
17.9 Summary
18 Applications Using One Independent Variable
18.1 Introduction
18.2 Word Problems
18.3 Review of Applications
18.4 Arc Length
18.5 Curvature Again
18.6 Surfaces of Rotation
18.7 Extensions
18.8 Derivative of an Integral
18.9* Mechanics
18.10* Force and Work
18.11* Inverse Square Law of Force
18.12 Summary
19 Applications Using Several Independent Variables
19.1 Fundamental Integral
19.2 Finding Volumes
19.3 Polar Coordinates
19.4 The Calculus in Polar Coordinates
19.5* The Distribution of Products of Random Numbers
19.6 The Jacobian
19.7 Three Independent Variables
19.8 Other Coordinate Systems
19.9 n-Dimensional Space
19.10 Parametric Equations
19.11 The Cycloid—Moving Coordinate Systems
19.12 Arc Length
19.13 Summary
IV - Miscellaneous Topics
20 Infinite Series
20.1 Review
20.2 Monotone Sequences
20.3 The Integral Test
20.4* Summation by Parts
20.5 Conditionally Convergent Series
20.6 Power Series
20.7 Maclaurin and Taylor Series
20.8 Some Common Power Series
20.9 Summary
21 Applications of Infinite Series
21.1 The Formal Algebra of Power Series
21.2 Generating Functions
21.3 The Binomial Expansion Again
21.4 Exponential Generating Functions
21.5 Complex Numbers Again
21.6 Hyperbolic Functions
21.7* Hyperbolic Functions Continued
21.8 Summary
22* Fourier Series
22.1 Introduction
22.2 Orthogonality
22.3 The Formal Expansion
22.4 Complex Fourier Series
22.5 Orthogonality and Least Squares
22.6 Convergence at a Point of Continuity
22.7 Convergence at a Point of Discontinuity
22.8 Rate of Convergence
22.9 Gibbs Phenomenon
22.10 The Finite Fourier Series
22.11 Summary
23 Differential Equations
23.1 What Is a Differential Equation?
23.2 What Is a Solution?
23.3 Why Study Differential Equations?
23.4 The Method of Variables Separable
23.5 Homogeneous Equations
23.6 Integrating Factors
23.7 First-Order Linear Differential Equations
23.8 Change of Variables
23.9 Special Second-Order Linear Differential Equations
23.10 Difference Equations
23.11 Summary
24 Linear Differential Equations
24.1 Introduction
24.2 Second-Order Equations with Constant Coefficients
24.3 The Nonhomogeneous Equation
24.4 Variation of Parameters Method
24.5 nth-Order Linear Equations
24.6 Equations with Variable Coefficients
24.7 Systems of Equations
24.8 Difference Equations
24.9 Summary
25 Numerical Methods
25.1 Roundoff and Truncation Errors
25.2 Analytic Substitution
25.3 Polynomial Approximation
25.4 The Direct Method
25.5 Least Squares
25.6 On Finding Formulas
25.7 Integration of Ordinary Differential Equations
25.8 Fourier Series and Power Series
25.9 Summary
26 Epilogue
26.1 Methods
26.2 Methods of Mathematics
26.3 Applications
26.4 Philosophy
Appendix A: Table of Integrals
Appendix B: Some Geometric Formulas
Appendix C: The Greek Alphabet
Answers to Some of the Exercises
Index
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