Foundations of Mathematics · Algebra, Geometry, Trigonometry, Calculus by Brown, Philip -- Read -- Imperial Library of Trantor

Index

Cover Page Title Copyright Dedication Contents Preface Acknowledgments 1 The Laws of Algebra

1.1 Introduction 1.2 Numbers 1.3 Fractions and Inequalities 1.4 Notation for Sets 1.5 Intervals of the Real Line and Set Intersections 1.6 Absolute Value of a Real Number 1.7 Algebraic Operations

1.7.1 Addition and Subtraction 1.7.2 Multiplication 1.7.3 Division 1.7.4 Recurring Decimals and Irrational Numbers

1.8 An Algebraic System on the Set of Real Numbers

1.8.1 Foil

1.9 Natural Numbers as Exponents 1.10 Order of Operations 1.11 Laws of Division 1.12 Decimal Notation

1.12.1 Decimal Representation of Fractions 1.12.2 Scientific Notation and Precision 1.12.3 Decimal Representation of Irrational Numbers 1.12.4 Conversion of Decimals into Fractions 1.12.5 Rounding Off Decimals

1.13 Divisibility of Natural Numbers

1.13.1 Prime Decomposition of a Natural Number 1.13.2 Finding the Prime Factors of a Natural Number 1.13.3 Testing for Divisibility by Small Prime Numbers 1.13.4 Adding Fractions Using Their Lowest Common Denominator

1.14 Laws for Exponents 1.15 Radicals Exercises

2 The Cartesian Plane

2.1 Introduction 2.2 Working in a Coordinate System 2.3 Linear Equations and Straight Lines

2.3.1 The Graph of a Linear Equation 2.3.2 The Linear Equation of a Line 2.3.3 Linear Relationships in Statistical Analysis 2.3.4 Parallel and Perpendicular Lines 2.3.5 The Distance between Points on a Line 2.3.6 The Equation of a Perpendicular Bisector

2.4 Circles in the Cartesian Plane 2.5 Conic Sections in the Cartesian Plane 2.6 Vector Algebra

2.6.1 Addition and Scalar Multiplication of Vectors 2.6.2 Subtraction of Vectors 2.6.3 The Standard Basis Vectors 2.6.4 The Dot Product of Vectors 2.6.5 The Triangle Inequality and the Parallelogram Law Exercises

3 Solving Equations and Factorizing Polynomials

3.1 Introduction 3.2 Solving Linear Equations 3.3 Solving Quadratic Equations by Completing the Square 3.4 Polynomials

3.4.1 Addition and Subtraction of Polynomials 3.4.2 Multiplication of Polynomials 3.4.3 Polynomials in More Than One Variable 3.4.4 Long Division of Polynomials 3.4.5 The Remainder Theorem and the Factor Theorem

3.5 The Properties of Quadratic Polynomials

3.5.1 The Graphs of Quadratic Polynomials 3.5.2 The Nature of the Roots 3.5.3 Factorizing Quadratic Polynomials 3.5.4 Solving Quadratic Equations by Factorizing

3.6 Complex Numbers as Matrices

3.6.1 The Algebra of × 2 2 Matrices 3.6.2 Complex Numbers as × 2 2 Matrices 743.7 Roots of Polynomials

3.7 Roots of Polynomials

3.7.1 Factorization Theorems 3.7.2 A Method For Finding the Integer and Rational Roots of a Polynomial

3.8 Graphs of Polynomials 3.9 Solving Cubic, Quartic, and Quintic Equations

3.9.1 Solving Cubic Equations 3.9.2 Solving Quartic Equations 3.9.3 Solving Quintic Equations Exercises

4 Trigonometry

4.1 Introduction 4.2 Angles in the Cartesian Plane 4.3 Trigonometric Ratios 4.4 Special Angles 4.5 Negative Angles and Periodicity 4.6 Reciprocal Trigonometric Ratios 4.7 Cofunction Identities 4.8 Trigonometric Graphs

4.8.1 Generation of a Sine Curve 4.8.2 Sine and Cosine Graphs 4.8.3 Scaling and Shifting of the Sine and Cosine Graphs 4.8.4 Tangent and Cotangent Graphs 4.8.5 Cosecant and Secant Graphs

4.9 Pythagorean Identities 4.10 Solving Basic Trigonometric Equations 4.11 Addition Identities 4.12 Double-Angle and Half-Angle Identities 4.13 Solving Triangles

4.13.1 Right Triangles 4.13.2 The Area Formula and the Sine Rule 4.13.3 The Cosine Rule

4.14 Vectors and Trigonometry

4.14.1 Components of a Vector 4.14.2 Geometric Interpretation of the Dot Product

4.15 More Identities Exercises

5 Functions

5.1 Introduction 5.2 Relations and Functions 5.3 Visualizing Functions

5.3.1 Graphs of Equations 5.3.2 The Vertical Line Test

5.4 The Absolute Value Function 5.5 Exponential Functions

5.5.1 Fractional Exponents 5.5.2 Irrational Exponents 5.5.3 The Graphs of Exponential Functions

5.6 Rational Functions 5.7 Root Functions 5.8 Piecewise Defined Functions 5.9 Symmetry of Functions 5.10 Operations on Functions

5.10.1 The Algebra of Functions 5.10.2 Compositions of Functions

5.11 Transformations of Functions

5.11.1 Vertical and Horizontal Shifts 5.11.2 Vertical and Horizontal Scaling 5.11.3 Reflections Across the Axes

5.12 Vector-Valued Functions

5.12.1 The Vector-Valued Function for a Circle 5.12.2 The Vector-Valued Function for a Line 5.12.3 Exploring Vector-Valued Functions

5.13 Inverse Functions

5.13.1 The Inverse of a Point 5.13.2 Logarithmic Functions 5.13.3 The Inversion of One-to-One Functions 5.13.4 Increasing and Decreasing Functions 5.13.5 Inverse Trigonometric Functions Exercises

6 Techniques of Algebra

6.1 Introduction 6.2 The Algebra of Rational Expressions

6.2.1 Multiplying and Dividing Rational Expressions 6.2.2 Adding and Subtracting Rational Expressions

6.3 Algebra with Rational Exponents 6.4 Solving Equations 6.5 Partial Fractions 6.6 Inequalities

6.6.1 Simplifying Inequalities 6.6.2 Solving Inequalities 6.6.3 Two-Variable Inequalities Exercises

7 Limits

7.1 Introduction 7.2 The Method of Exhaustion 7.3 Sequences 7.4 Limits of a Function 7.5 Continuity

7.5.1 Definition of Continuity at a Point 7.5.2 Discontinuity at a Point 7.5.3 Continuity on an Interval 7.5.4 Continuous Functions 7.5.5 More Continuous Functions

7.6 Computing Limits

7.6.1 Limits in the Domain of a Continuous Function 7.6.2 Limits Involving Piecewise-Defined Functions 7.6.3 Computing Limits by Simplification

7.7 Applications of Continuity

7.7.1 The Intermediate Value Theorem

7.8 Horizontal Asymptotes 7.9 Vertical Asymptotes of Rational Functions 7.10 The Squeeze Theorem and Rules for Limits

7.10.1 The Squeeze Theorem 7.10.2 The Rules for Limits Exercises

8 Differential Calculus

8.1 Introduction 8.2 Definition of the Derivative

8.2.1 Graphs Tangential to the x-axis at the Origin 8.2.2 The Tangent Line to a Graph at the Origin 8.2.3 A Formula for the Derivative 8.2.4 Definition of the Derivative (General Case)

8.3 Derivative Functions

8.3.1 The Power Rule for Natural Numbers 8.3.2 Leibniz Notation 8.3.3 The Sum, Product, and Quotient Rules

8.4 Tangent Line Problems 8.5 The Power Rule for Rational Exponents 8.6 Derivatives of Trigonometric Functions 8.7 Some Basic Applications of Calculus 8.8 The Chain Rule 8.9 The Calculus of Vector-Valued Functions 8.10 The Calculus of Exponential and Logarithmic Functions

8.10.1 A Formula for e 8.10.2 Derivatives of Exponential Functions 8.10.3 Derivatives of Logarithmic Functions 8.10.4 The Proof of the Power Rule

8.11 Derivatives of the Inverse Trigonometric Functions Exercises

9 Euclidean Geometry

9.1 Introduction 9.2 Euclid’s Elements 9.3 Terminology

9.3.1 Lines, Angles, and Polygons 9.3.2 Circles 9.3.3 Other Important Terms

9.4 Basic Problem Solving in Geometry 9.5 Elementary Theorems Relating to Lines and Polygons

9.5.1 Theorems about Angles 9.5.2 Theorems about Triangles 9.5.3 Parallelograms and Parallel Lines 9.5.4 Concurrency, Proportionality, and Similarity

9.6 Elementary Theorems Relating to Circles

9.6.1 Chords and Subtended Angles 9.6.2 Cyclic Quadrilaterals 9.6.3 Tangent Lines and Secant Lines

9.7 Examples and Applications Exercises Chapter Appendix: Geometry Theorems

10 Spherical Trigonometry

10.1 Introduction 10.2 Planes and Spheres

10.2.1 Planes in Space 10.2.2 Spheres 10.2.3 Spherical Triangles

10.3 Vectors in Space

10.3.1 The Cross Product of Two Vectors 10.3.2 Parallelepipeds and Cross Product Identities 10.3.2 The Angle Between Two Planes

10.4 Solving Spherical Triangles 10.5 Solving Right Spherical Triangles (I) 10.6 Solving Right Spherical Triangles (II)

10.6.1 Rules for Quadrants 10.6.2 Napier’s Rules

Exercises

Appendix A: Answers to Selected Exercises Index

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