Precalculus: With Limits, First Edition
January 2010, ©2010
Precalculus is suited for a 1 or 2 semester course and will focus on different learning styles. There will also be an emphasis on more challenging problems as well as streamlined prose to quicken the pace of the book.
0.1 Linear Equations
0.2 Quadratic Equations
0.3 Other Types of Equations
0.5 Graphing Equations
0.7 Modeling Variation
1 Functions and Their Graphs
1.2 Graphs of Functions
1.3 Graphing Techniques: Transformations
1.4 Combining Functions
1.5 One-to-One Functions and Inverse Functions
2 Polynomial and Rational Functions
2.1 Quadratic Functions
2.2 Polynomial Functions of Higher Degree
2.3 Dividing Polynomials
2.4 The Real Zeros of a Polynomial Function
2.5 Complex Zeros: The Fundamental Theorem of Algebra
2.6 Rational Functions
3 Exponential and Logarithmic Functions
3.1 Exponential Functions and Their Graphs
3.2 Logarithmic Functions and Their Graphs
3.3 Properties of Logarithms
3.4 Exponential and Logarithmic Equations
3.5 Exponential and Logarithmic Models
4 Trigonometric Functions of Angles
4.1 Angle Measure
4.2 Right Triangle Trigonometry
4.3 Trigonometric Functions of Angles
4.4 The Law of Sines
4.5 The Law of Cosines
5 Trigonometric Functions of Real Numbers
5.1 Trigonometric Functions: The Unit Circle Approach
5.2 Graphs of Sine and Cosine Functions
5.3 Graphs of Other Trigonometric Functions
6 Analytic Trigonometry
6.1 Verifying Trigonometric Identities
6.2 Sum and Difference Identities
6.3 Double-Angle and Half-Angle Identities
6.4 Product-to-Sum and Sum-to-Product Identities
6.5 Inverse Trigonometric Functions
6.6 Trigonometric Equations
7 Vectors, the Complex Plane, and Polar Coordinates
7.2 The Dot Product
7.3 Polar (Trigonometric) Form of Complex Numbers
7.4 Products, Quotients, Powers, and Roots of Complex Numbers
7.5 Polar Coordinates and Graphs of Polar Equations
8 Systems of Linear Equations and Inequalities
8.1 Systems of Linear Equations in Two Variables
8.2 Systems of Linear Equations in Three Variables
8.3 Systems of Linear Equations and Matrices
8.4 Matrix Algebra
8.5 The Determinant of a Square Matrix and Cramer’s Rule
8.6 Partial Fractions
8.7 Systems of Linear Inequalities in Two Variables
9 Conics, Systems of Nonlinear Equations and Inequalities, and Parametric Equations
9.1 Conic Basics
9.2 The Parabola
9.3 The Ellipse
9.4 The Hyperbola
9.5 Systems of Nonlinear Equations
9.6 Systems of Nonlinear Inequalities
9.7 Rotation of Axes
9.8 Polar Equations of Conics
9.9 Parametric Equations and Graphs
10 Sequences and Series
10.1 Sequences and Series
10.2 Arithmetic Sequences and Series
10.3 Geometric Sequences and Series
10.4 Mathematical Induction
10.5 The Binomial Theorem
11 Limits: A Preview to Calculus 992
11.1 Introduction to Limits: Estimating Limits Numerically and Graphically
11.2 Techniques for Finding Limits
11.3 Tangent Lines and Derivatives
11.4 Limits at Infinity; Limits of Sequences
11.5 Finding the Area Under a Curve
Appendix Prerequisites: Fundamentals of Algebra
A.1 Real Numbers
A.2 Integer Exponents and Scientific Notation
A.3 Polynomials: Basic Operations
A.4 Factoring Polynomials
A.5 Rational Expressions
A.6 Rational Exponents and Radicals
A.7 Complex Numbers
Chapter Opening Vignette: Peaks the students interest with a real-world application of material presented in that chapter.
Chapter Overview/Flowchart: Students see the big picture of how topics relate to each other.
Skills and Conceptual Objectives: Emphasize conceptual understanding and push students to think at a more global level on the topics presented.
Clear, Concise, and Inviting Writing Style, Tone, and Layout: Students are able to read this book, which reduces math anxiety and encourages student success.
Parallel Words and Math: Increases students’ ability to read and understand examples.
Common Mistake/ Correct vs. Incorrect Boxes: Demonstrates common mistakes so students understand why a step is incorrect.
Color for Pedagogical Reasons: A student sees that a function is written in red in the text, then its graph is also red, whereas another function is written in blue and its graph is blue.
Study Tips: Serves to reinforce notes you would write on the board in class.
Technology Tips: Demonstrates use of technology to confirm analytic solutions.
Your Turn: Engages students during class, build student confidence, and assist instructor in assessing class understanding
Catch the Mistake Exercises: Encourages students to take the role of teacher.
Conceptual Exercises: Students have to answer questions about all types of functions, not a specific function.
Preview to Calculus: Demonstrate where the current precalculus topics will be used in
calculus. This helps answer the question “when will I use this?”
Modeling Your World: Engages students through timely issues. When students use mathematical models they develop as a result of this course, they value the course more.
Chapter Review (Key Ideas:) Key formulas and ideas are presented section by section in a chart formatimproves study skills.
Chapter Review Exercises: Improves study skills.
Chapter Practice Test: Offers self-assessment and improves study skills.
Cumulative Test: Improves retention.