front cover -- copyright page -- title page -- Contents -- Preface -- Overview -- Mathematical Content and Organization -- Pedagogical Philosophy -- Audience, Level, and Prerequisites -- Features of this Text -- History of this Text -- To the Student -- To the Instructor -- Instructor Not Included -- Sample Courses -- Sequencing Chapters in a Course -- Special Topics -- Approaches to Basic Material -- Use of Let's Go and Your Turn Activities -- Use of Exercises -- Analyzing Student Work -- 1 Functions -- 1.1 The Notion of a Function -- 1.1.1 Seeking a definition -- 1.1.2 Why sets? -- 1.1.3 What the definition says, and what it doesn't say -- 1.1.4 Exercises -- 1.2 Graphs, Relations, and the Definition of a Function -- 1.2.1 The graph of a function -- 1.2.2 Functions, rigorously defined -- 1.2.3 Functional notation -- 1.2.4 Exercises -- 1.3 An Application/Interlude: Graphing Transformations -- 1.3.1 An example: a horizontal shift of a graph -- 1.3.2 Exercises -- 1.4 Images and Inverse Images -- 1.4.1 Images of points and sets -- 1.4.2 Inverse images of sets -- 1.4.3 Exercises -- 1.5 Injective, Surjective, and Bijective Functions -- 1.5.1 Counting points in inverse images -- 1.5.2 Injective functions -- 1.5.3 Surjective functions -- 1.5.4 Bijective functions -- 1.5.5 Exercises -- 1.6 Composite Functions, Identity Functions, and Inverse Functions -- 1.6.1 Composite functions -- 1.6.2 The identity function -- 1.6.3 Inverses of Functions -- 1.6.4 Exercises -- 1.7 Subsets and Equations -- 1.7.1 Two methods of describing a subset -- 1.7.2 Solution Sets -- 1.7.3 Exercises -- 2 Lines in the Plane -- 2.1 Linear Equations -- 2.1.1 Algebraic and geometric interpretations of lines -- 2.1.2 Exercises -- 2.2 Intercepts, Slope, and Equations of Lines -- 2.2.1 Slopes of lines -- 2.2.2 Intercepts -- 2.2.3 Forms for the equation of a line -- 2.2.4 Exercises.

2.3 Solving Systems of Linear Equations -- 2.3.1 Elimination -- 2.3.2 Substitution -- 2.3.3 Exercises -- 2.4 Parametrized Lines -- 2.4.1 Whirlwind review of vectors -- 2.4.2 Moving with constant velocity: parametrizations of lines -- 2.4.3 Different parametrizations for the same line -- 2.4.4 Intersecting parametrized lines -- 2.4.5 Converting between parametric and non-parametric equations -- 2.4.6 Exercises -- 2.5 Parallel Lines, Perpendicular Lines, and Distance -- 2.5.1 Perpendicular vectors -- 2.5.2 "Perpendicular form" for the equation of a line -- 2.5.3 Parallel lines, perpendicular lines, and familiar formulas -- 2.5.4 The distance between parallel lines -- the distance between a line and a point -- 2.5.5 Exercises -- 2.6 An Application of Distance: Using Lines to Fit Data -- 2.6.1 Pipelines and perpendicular offsets -- 2.6.2 Vertical offsets, magnetic fields, and linear regression -- 2.6.3 Exercises -- 2.7 The Collection of All Lines in the Plane -- 2.7.1 All non-vertical lines and the mb-plane -- 2.7.2 Non-vertical, non-horizontal lines and the ab-plane -- 2.7.3 The collection of all lines in the xy plane -- 2.7.4 Exercises -- 2.8 Analysis of Student Work -- 3 Quadratic Polynomials -- 3.1 The Squaring Function and Parabolas -- 3.1.1 The graph of the squaring function: more than just a U-shaped curve -- 3.1.2 The reflection property -- 3.1.3 Exercises -- 3.2 Completing the Square and Solving Quadratic Equations -- 3.2.1 Algebraic viewpoint -- 3.2.2 Geometric viewpoint -- 3.2.3 Exercises -- 3.3 Quadratic Polynomials: Graphs, Roots, and Factors -- 3.3.1 Graphing transformations and parabolas -- 3.3.2 Quadratic formula, the nature of roots, and the bc-plane -- 3.3.3 Factoring quadratic polynomials -- 3.3.4 Exercises -- 3.4 Application: Quadratic Splines -- 3.4.1 Fitting a quadratic to three points.

3.4.2 Fitting quadratics to more points: Quadratic Splines -- 3.4.3 Exercises -- 3.5 Application: Problems Involving Tangency -- 3.5.1 A circle tangent to a parabola -- 3.5.2 Dual circles -- 3.5.3 Exercises -- 3.6 Analysis of Student Work -- 4 Trigonometry -- 4.1 Angles -- 4.1.1 The meaning of angle: an exploration -- 4.1.2 Settling on an interpretation of the word "angle" -- 4.1.3 Angle measurement -- 4.1.4 Exercises -- 4.2 Triangles, Circles, and Trigonometric Functions -- 4.2.1 Trigonometric functions and triangles -- 4.2.2 Trigonometric functions and circles -- 4.2.3 Graphs and periodicity -- 4.2.4 Inverse trigonometric functions -- 4.2.5 Exercises -- 4.3 Laws of Sines and Cosines -- 4.3.1 The Laws of Sines and Cosines -- 4.3.2 Exercises -- 4.4 Values of the Trigonometric Functions: Ptolemy's Theorem,Identities, and Tables -- 4.4.1 Ptolemy's Theorem -- 4.4.2 Useful identities -- 4.4.3 Trigonometric tables and the problem of tedious multiplication -- 4.4.4 Exercises -- 4.5 Trigonometry, Coordinate Geometry, and Linear Algebra -- 4.5.1 The sum formulas revisited -- 4.5.2 The rotation matrix and rotated coordinate systems -- 4.5.3 Complex multiplication -- 4.5.4 Exercises -- 4.6 Application: Why We Use Radians in Calculus -- 4.6.1 Exercises -- 4.7 Application: Periodic Data and Trigonometric Polynomials -- 4.7.1 Trigonometric polynomials and data -- 4.7.2 A least squares minimization problem and orthogonality -- 4.7.3 A coding scheme: put a wave in the air -- 4.7.4 Verifying the orthogonality relations: the triumph of De Moivre's formula -- 4.7.5 Exercises -- 4.8 Analysis of Student Work -- 5 Hyperbolic Trigonometry -- 5.1 Hyperbolas and Ellipses -- 5.1.1 Ellipses -- 5.1.2 Hyperbolas -- 5.1.3 Special ellipses and hyperbolas -- 5.1.4 Exercises -- 5.2 Hyperbolic Angles and Their Measures -- 5.2.1 Hyperbolic angles -- 5.2.2 Measure of hyperbolic angles.

5.2.3 Producing a standard angle from a number -- 5.2.4 Exercises -- 5.3 The Hyperbolic Trigonometric Functions -- 5.3.1 Circular and hyperbolic trigonometric functions -- 5.3.2 Exercises -- 5.4 Hyperbolic Rotations -- 5.4.1 What should a rotation do? -- 5.4.2 Rotations as linear transformations -- 5.4.3 A proof of Theorem 2 -- 5.4.4 Exercises -- 5.5 Differential Calculus and the Hyperbolic Trigonometric Functions -- 5.5.1 The derivative of the hyperbolic sine and cosine functions -- 5.5.2 Exercises -- 5.6 Hyperbolic Trigonometric Functions and the Exponential Function -- 5.6.1 Calculus to the rescue -- 5.6.2 Exercises -- 5.7 Linking the Two Trigonometries -- 5.7.1 Circles and hyperbolas together -- 5.7.2 Parametrizations of the hyperbola, and the Gudermannian function -- 5.7.3 Exercises -- 6 Numbers -- 6.1 The Natural Numbers and Whole Numbers: An Intuitive View -- 6.1.1 Numbers and counting -- 6.1.2 Exercises -- 6.2 The Integers: An Intuitive View -- 6.2.1 The need for negative numbers -- 6.2.2 Exercises -- 6.3 The Rational Numbers: An Intuitive View -- 6.3.1 Parts of a whole -- 6.3.2 Exercises -- 6.4 The Real Numbers: An Intuitive View -- 6.4.1 The number line -- 6.4.2 Absolute value -- 6.4.3 Exercises -- 6.5 The Natural Numbers and Whole Numbers: A More Rigorous View -- 6.5.1 A confession about set theory -- 6.5.2 Finding the next biggest number: the successor function -- 6.5.3 When do two sets have the same number of elements? -- 6.5.4 Comparing the size of two sets -- 6.5.5 Finite and infinite sets -- 6.5.6 Equivalence classes of finite sets? -- 6.5.7 Exercises -- 6.6 The Integers: A More Rigorous View -- 6.6.1 Defining the integers -- 6.6.2 Exercises -- 6.7 The Rational Numbers: A More Rigorous View -- 6.7.1 Fractions and rational numbers -- 6.7.2 Exercises -- 6.8 The Real Numbers: A More Rigorous View -- 6.8.1 A formal definition.

6.8.2 Comparing real numbers -- 6.8.3 No holes in the reals -- 6.8.4 Exercises -- 6.9 Analysis of Student Work -- 7 Operations in Number Systems -- 7.1 Addition and Subtraction in Number Systems -- 7.1.1 Intuitive models -- 7.1.2 A more rigorous viewpoint -- 7.1.3 Exercises -- 7.2 Multiplication in Number Systems -- 7.2.1 Whole number multiplication -- 7.2.2 Positive rational numbers -- 7.2.3 Multiplication with negative numbers -- 7.2.4 Exercises -- 7.3 Division in Number Systems -- 7.3.1 A definition of division -- 7.3.2 Division and multiplicative inverses -- 7.3.3 Models of division -- 7.3.4 Exercises -- 7.4 Algebraic Properties in Number Systems: An Intuitive View -- 7.4.1 Properties of addition and multiplication in the real numbers -- 7.4.2 Exercises -- 7.5 Algebraic Properties in Number Systems: A Survey of Some RigorousArguments -- 7.5.1 Whole numbers -- 7.5.2 Integers -- 7.5.3 Rational numbers -- 7.5.4 Real numbers -- 7.5.5 Exercises -- 7.6 Complex Numbers -- 7.6.1 The complex numbers: set and operations -- 7.6.2 Algebraic properties -- 7.6.3 Polar decomposition -- 7.6.4 The geometric meaning of complex multiplication -- 7.6.5 Why complex numbers? -- 7.6.6 Exercises -- 8 Topics in Number Systems -- 8.1 Arithmetic in the Integers -- 8.1.1 Divisors and multiples -- 8.1.2 Greatest common divisors and least common multiples -- 8.1.3 Primes -- 8.1.4 The Fundamental Theorem of Arithmetic -- 8.1.5 Applications of FTA -- 8.1.6 The Division Algorithm -- 8.1.7 Exercises -- 8.2 Systems of Numeration for Whole Numbers -- 8.2.1 Numbers and numerals -- 8.2.2 Ancient numeration systems -- 8.2.3 Hindu-Arabic numeration -- 8.2.4 Exercises -- 8.3 Divisibility Tests -- 8.3.1 Familiar tests and their proofs -- 8.3.2 Exercises -- 8.4 Decimals -- 8.4.1 Making sense of decimals -- 8.4.2 Some technicalities.

8.4.3 Rational versus irrational numbers and their decimal expansions.