What is included with this book?
About the Author | |
Preface to the Instructor | |
Acknowledgments | |
Preface to the Student | |
The Real Numbers | |
The Real Line | |
Construction of the Real Line | |
Is Every Real Number Rational? | |
Problems | |
Algebra of the Real Numbers | |
Commutativity and Associativity | |
The Order of Algebraic Operations | |
The Distributive Property | |
Additive Inverses and Subtraction | |
Multiplicative Inverses and the Algebra of Fractions | |
Symbolic Calculators | |
Exercises, Problems, and Worked-out Solutions | |
Inequalities | |
Positive and Negative Numbers | |
Lesser and Greater | |
Intervals | |
Absolute Value | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
Combining Algebra and Geometry | |
The Coordinate Plane | |
Coordinates | |
Graphs of Equations | |
Distance Between Two Points | |
Length, Perimeter, and Circumference | |
Exercises, Problems, and Worked-out Solutions | |
Lines | |
Slope | |
The Equation of a Line | |
Parallel Lines | |
Perpendicular Lines | |
Midpoints | |
Exercises, Problems, and Worked-out Solutions | |
Quadratic Expressions and Conic Sections | |
Completing the Square | |
The Quadratic Formula | |
Circles | |
Ellipses | |
Parabolas | |
Hyperbolas | |
Exercises, Problems, and Worked-out Solutions | |
Area | |
Squares, Rectangles, and Parallelograms | |
Triangles and Trapezoids | |
Stretching | |
Circles and Ellipses | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
Functions and Their Graphs | |
Functions | |
Definition and Examples | |
The Graph of a Function | |
The Domain of a Function | |
The Range of a Function | |
Functions via Tables | |
Exercises, Problems, and Worked-out Solutions | |
Function Transformations and Graphs | |
Vertical Transformations: Shifting, Stretching, and Flipping | |
Horizontal Transformations: Shifting, Stretching, Flipping | |
Combinations of Vertical Function Transformations | |
Even Functions | |
Odd Functions | |
Exercises, Problems, and Worked-out Solutions | |
Composition of Functions | |
Combining Two Functions | |
Definition of Composition | |
Order Matters in Composition | |
Decomposing Functions | |
Composing More than Two Functions | |
Function Transformations as Compositions | |
Exercises, Problems, and Worked-out Solutions | |
Inverse Functions | |
The Inverse Problem | |
One-to-one Functions | |
The Definition of an Inverse Function | |
The Domain and Range of an Inverse Function | |
The Composition of a Function and Its Inverse | |
Comments about Notation | |
Exercises, Problems, and Worked-out Solutions | |
A Graphical Approach to Inverse Functions | |
The Graph of an Inverse Function | |
Graphical Interpretation of One-to-One | |
Increasing and Decreasing Functions | |
Inverse Functions via Tables | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
Polynomial and Rational Functions | |
Integer Exponents | |
Positive Integer Exponents | |
Properties of Exponents | |
Defining x0 | |
Negative Integer Exponents | |
Manipulations with Exponents | |
Exercises, Problems, and Worked-out Solutions | |
Polynomials | |
The Degree of a Polynomial | |
The Algebra of Polynomials | |
Zeros and Factorization of Polynomials | |
The Behavior of a Polynomial Near -1 | |
Graphs of Polynomials | |
Exercises, Problems, and Worked-out Solutions | |
Rational Functions | |
Ratios of Polynomials | |
The Algebra of Rational Functions | |
Division of Polynomials | |
The Behavior of a Rational Function Near -1 | |
Graphs of Rational Functions | |
Exercises, Problems, and Worked-out Solutions | |
Complex Numbers | |
The Complex Number System | |
Arithmetic with Complex Numbers | |
Complex Conjugates and Division of Complex Numbers | |
Zeros and Factorization of Polynomials, Revisited | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
Exponents and Logarithms | |
Exponents and Exponential Functions | |
Roots | |
Rational Exponents | |
Real Exponents | |
Exponential Functions | |
Exercises, Problems, and Worked-out Solutions | |
Logarithms as Inverses of Exponential Functions | |
Logarithms Base 2 | |
Logarithms with Any Base | |
Common Logarithms and the Number of Digits | |
Logarithm of a Power | |
Radioactive Decay and Half-Life | |
Exercises, Problems, and Worked-out Solutions | |
Applications of Logarithms | |
Logarithm of a Product | |
Logarithm of a Quotient | |
Earthquakes and the Richter Scale | |
Sound Intensity and Decibels | |
Star Brightness and Apparent Magnitude | |
Change of Base | |
Exercises, Problems, and Worked-out Solutions | |
Exponential Growth | |
Functions with Exponential Growth | |
Population Growth | |
Compound Interest | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
e and the Natural Logarithm | |
Defining e and ln | |
Estimating Area Using Rectangles | |
Defining e | |
Defining the Natural Logarithm | |
Properties of the Exponential Function and ln | |
Exercises, Problems, and Worked-out Solutions | |
Approximations with e and ln | |
Approximation of the Natural Logarithm | |
Inequalities with the Natural Logarithm | |
Approximations with the Exponential Function | |
An Area Formula | |
Exercises, Problems, and Worked-out Solutions | |
Exponential Growth Revisited | |
Continuously Compounded Interest | |
Continuous Growth Rates | |
Doubling Your Money | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
Trigonometric Functions | |
The Unit Circle | |
The Equation of the Unit Circle | |
Angles in the Unit Circle | |
Negative Angles | |
Angles Greater Than 360- | |
Length of a Circular Arc | |
Special Points on the Unit Circle | |
Exercises, Problems, and Worked-out Solutions | |
Radians | |
A Natural Unit of Measurement for Angles | |
Negative Angles | |
Angles Greater Than 2- | |
Length of a Circular Arc | |
Area of a Slice | |
Special Points on the Unit Circle | |
Exercises, Problems, and Worked-out Solutions | |
Cosine and Sine | |
Definition of Cosine and Sine | |
Cosine and Sine of Special Angles | |
The Signs of Cosine and Sine | |
The Key Equation Connecting Cosine and Sine | |
The Graphs of Cosine and Sine | |
Exercises, Problems, and Worked-out Solutions | |
More Trigonometric Functions | |
Definition of Tangent | |
Tangent of Special Angles | |
The Sign of Tangent | |
Connections between Cosine, Sine, and Tangent | |
The Graph of Tangent | |
Three More Trigonometric Functions | |
Exercises, Problems, and Worked-out Solutions | |
Trigonometry in Right Triangles | |
Trigonometric Functions via Right Triangles | |
Two Sides of a Right Triangle | |
One Side and One Angle of a Right Triangle | |
Exercises, Problems, and Worked-out Solutions | |
Trigonometric Identities | |
The Relationship Between Cosine and Sine | |
Trigonometric Identities for the Negative of an Angle | |
Trigonometric Identities with | |
Trigonometric Identities Involving a Multiple of | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
Trigonometric Algebra and Geometry | |
Inverse Trigonometric Functions | |
The Arccosine Function | |
The Arcsine Function | |
The Arctangent Function | |
Exercises, Problems, and Worked-out Solutions | |
Inverse Trigonometric Identities | |
The Arccosine, Arcsine, and Arctangent of | |
t: Graphical | |
Approach | |
The Arccosine, Arcsine, and Arctangent of | |
t: Algebraic | |
Approach | |
Arccosine Plus Arcsine | |
The Arctangent of 1t | |
Composition of Trigonometric Functions and Their Inverses | |
More Compositions with Inverse Trigonometric Functions | |
Exercises, Problems, and Worked-out Solutions | |
Using Trigonometry to Compute Area | |
The Area of a Triangle via Trigonometry | |
Ambiguous Angles | |
The Area of a Parallelogram via Trigonometry | |
The Area of a Polygon | |
Exercises, Problems, and Worked-out Solutions | |
The Law of Sines and the Law of Cosines | |
The Law of Sines | |
Using the Law of Sines | |
The Law of Cosines | |
Using the Law of Cosines | |
When to Use Which Law | |
Exercises, Problems, and Worked-out Solutions | |
Double-Angle and Half-Angle Formulas | |
The Cosine of 2- | |
The Sine of 2- | |
The Tangent of 2- | |
The Cosine and Sine of -2 | |
The Tangent of -2 | |
Exercises, Problems, and Worked-out Solutions | |
Addition and Subtraction Formulas | |
The Cosine of a Sum and Difference | |
The Sine of a Sum and Difference | |
The Tangent of a Sum and Difference | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
Applications of Trigonometry | |
Parametric Curves | |
Curves in the Coordinate Plane | |
Graphing Inverse Functions as Parametric Curves | |
Shifting, Stretching, or Flipping a Parametric Curve | |
Exercises, Problems, and Worked-out Solutions | |
Transformations of Trigonometric Functions | |
Amplitude | |
Period | |
Phase Shift | |
Exercises, Problems, and Worked-out Solutions | |
Polar Coordinates | |
Defining Polar Coordinates | |
Converting from Polar to Rectangular Coordinates | |
Converting from Rectangular to Polar Coordinates | |
Graphs of Polar Equations | |
Exercises, Problems, and Worked-out Solutions | |
Vectors | |
An Algebraic and Geometric Introduction to Vectors | |
Vector Addition | |
Vector Subtraction | |
The Dot Product | |
Exercises, Problems, and Worked-out Solutions | |
The Complex Plane | |
Complex Numbers as Points in the Plane | |
Geometric Interpretation of Complex Multiplication and Division | |
De Moivre's Theorem | |
Finding Complex Roots | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
Systems of Equations and Inequalities | |
Equations and Systems of Equations | |
Solving an Equation | |
Solving a System of Equations | |
Systems of Linear Equations | |
Matrices | |
Exercises, Problems, and Worked-out Solutions | |
Solving Systems of Linear Equations | |
Gaussian Elimination | |
Gaussian Elimination with Matrices | |
Special Cases-No Solutions | |
Special Cases-Infinitely Many Solutions | |
Exercises, Problems, and Worked-out Solutions | |
Matrix Algebra | |
Adding and Subtracting Matrices | |
Multiplying Matrices | |
The Inverse of a Matrix | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
Sequences, Series, and Limits | |
Sequences | |
Introduction to Sequences | |
Arithmetic Sequences | |
Geometric Sequences | |
Recursively-Defined Sequences | |
Exercises, Problems, and Worked-out Solutions | |
Series | |
Sums of Sequences | |
Arithmetic Series | |
Geometric Series | |
Summation Notation | |
The Binomial Theorem | |
Exercises, Problems, and Worked-out Solutions | |
Limits | |
Introduction to Limits | |
Infinite Series | |
Decimals as Infinite Series | |
Special Infinite Series | |
Exercises, Problems, and Worked-out Solutions | |
Chapter Summary and Chapter Review Questions | |
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