Fundamentals of Freshman Mathematics

Chapter 1. Mathematics and Science
1.1. Introduction
1.2. Abstract Nature of Mathematics
1.3. Negations
1.4. Implications
1.5. Necessary and Sufficient Conditions
1.6. Direct Proof
1.7. Other Methods of Proof
1.8. Methods of Disproof
1.9. Mathematical Models

Chapter 2. The Number System
2.1. Introduction
2.2. Addition of Real Numbers
2.3. Multiplication of Real Numbers
2.4. Formal Properties of Real Numbers
2.5. Special Properties of the Natural Numbers - Mathematical Induction
2.6. Special Properties of Zero
2.7. Special Properties of the Integers
2.8. Special Properties of the Rational Numbers
2.9. Decimal Expansions 
2.10. Some Irrational Numbers
2.11. Geometric Representation of Real Numbers
2.12. The Use of Real Numbers in the Plane
2.13. Lengths of Segments; Units on the Axes
2.14. Complex Numbers
2.15. Solutions of Other Algebraic Equations
2.16. Classification of Numbers

Chapter 3. Polynomials
3.1. Algebraic Expressions
3.2. Addition of Polynomials
3.3. Multiplication of Polynomials
3.4. Binomial Theorem
3.5. Division of Polynomials
3.6. Factoring

Chapter 4. Algebraic Fractions
4.1. Introduction
4.2. Simplification of Fractions
4.3. Addition
4.4. Multiplication and Division
4.5. Compound  Fractions

Chapter 5. Exponents and Radicals
5.1. Positive Integral Exponents
5.2. Negative and Zero Exponents
5.3. Fractional Exponents
5.4. Special Problems Concerning Square Roots
5.5. Special Problems Concerning Odd Roots
5.6. Unanswered Questions
5.7. Rationalizing Denominators

Chapter 6. Sets and Equations
6.1. Sets
6.2. Subsets
6.3. Union and Intersection
6.4. Sets Defined by Equations
6.5. Linear Equations
6.6. Quadratic Equations
6.7. Equations Containing Fractions
6.8. Equations Containing Radicals

Chapter 7. Simultaneous Equations and Matrices
7.1. Linear Equations and Their Graphs
7.2. The Graph of a Set of Ordered Pairs
7.3. Simultaneous Linear Equations
7.4. Simultaneous Linear Equations (Continued)
7.5. Simultaneous Linear Equations in Three Unknowns
7.6. Vectors
7.7. Products of Vectors
7.8. Matrices
7.9. Products of Matrices
7.10. Inverse of a Square Matrix
7.11. Determinants
7.12. Applications of Matrices to Simultaneous Equations
7.13. Word Problems

Chapter 8. Inequalities
8.1. Introduction
8.2. Theorems about Inequalities
8.3. Linear Inequalities
8.4. Quadratic Inequalities
8.5. The Graph of a Linear Inequality
8.6. Simultaneous Linear Inequalities
8.7. Applications

Chapter 9. Functions and Relations
9.1. Relations
9.2. Functions
9.3. Absolute-value Function
9.4. Algebra of Functions
9.5. Graphs
9.6. Graphs (Continued)
9.7. Inverse Functions
9.8. Functions Derived from Equations

Chapter 10. Algebraic Functions
10.1. Introduction
10.2. Polynomial Functions
10.3. Rational Functions
10.4. Explicit Algebraic Functions
10.5. Graphs and Continuity
10.6. Properties of Polynomials
10.7. Synthetic Division
10.8. Roots of Polynomial Equations
10.9. Rational Roots of Rational Polynomial Equations
10.10. Real Roots of Real Polynomial Equations

Chapter 11. Exponential and Logarithmic Functions
11.1. Exponential Functions
11.2. The Number e
11.3. Logarithmic Functions
11.4. Graphs
11.5. Applications
11.6. The Logarithmic Scale

Chapter 12. Trigonometric Functions of Angles
12.1. Introduction
12.2. Distance in the Plane
12.3. Directed Angles
12.4. Polar Coordinates
12.5. Sine and Cosine of a Directed Angle
12.6. Sine and Cosine of Special Angles
12.7. Other Trigonometric Functions
12.8. Some Important Identities
12.9. Trigonometric Tables
12.10. Right Triangles
12.11. Vectors
12.12. Law of Sines
12.13. Law of Cosines
12.14. Law of Tangents

13. Trigonometric Functions of Real Numbers
13.1. Arc Length and Radian Measure
13.2. Computations
13.3. Range and Graphs of the Functions
13.4. Amplitude, Period, Phase
13.5. Addition Theorems
13.6. Multiple- and Half-angle Formulas
13.7. Identities
13.8. Equations
13.9. Inverse Trigonometric Functions
13.10. Complex Numbers

Chapter 14. Analytic Geometry
14.1. Introduction
14.2. Mid-point of a Line Segment
14.3. Directed Line Segment
14.4. Rise, Run, Slope, Inclination
14.5. Direction Cosines
14.6. Angle between Two Directed Lines
14.7. Applications to Plane Geometry
14.8. The Straight Line
14.9. Conic Sections
14.10. Case I. The Circle
14.11. Case II. The Parabola
14.12. Case III. The Ellipse
14.13. Case IV. The Hyperbola
14.14. Applications
14.15. Polar Coordinates
14.16. Polar Coordinates (Continued)
14.17. Parametric Equations

Chapter 15. Intuitive Integration
15.1. Introduction
15.2. Area of a Circle
15.3. Some Limits
15.4. Area under y = x^2
15.5. Area under y  = x^n
15.6. Area under Graph of a Polynomial Function
15.7. Area under y = f(x)
15.8. Integration
15.9. Setting Up Problems; Applications

Chapter 16. Intuitive Differentiation
16.1. Introduction
16.2. Notion of a Tangent
16.3. Velocity and Acceleration
16.4. Derivative
16.5. Second Derivative
16.6. The Chain Rule
16.7. Maxima and Minima
16.8. Related Rates
16.9. Fundamental Theorem of Calculus
16.10. Falling Bodies

Chapter 17. Hyperbolic Functions
17.1. Hyperbolic Functions
17.2. Hyperbolic and Circular Trigonometric Functions
17.3. Hyperbolic Trigonometry
17.4. Euler's Formula

Answers

Index