Rockswold - Preface
Precalculus through Modeling and Visualization offers an innovative approach that consistently links mathematical concepts to real-world applications by moving from the concrete to the abstract. It demonstrates the relevance of mathematics and answers the question: "When will I ever need to know this?" This text provides a comprehensive curriculum with the balance and flexibility necessary for today's college algebra courses. The early introduction of functions and graphs allows the instructor to use applications and visualization to present mathematical topics. Real data, graphs, and tables play an important role in the course, giving meaning to the numbers and equations that students encounter. This approach increases student interest, motivation, and the likelihood for success.
Approach
The concept of a function is the unifying theme of the text. Functions and their graphs are frequently used to model data, with students often being asked to interpret their results. Mathematical skill building also plays an important role throughout the text. Instructors are given the freedom to strike their own balance with regard to emphasis on skills, rule of four, applications, modeling, and technology. With a flexible approach to the rule of four (verbal, graphical, numerical, and symbolic methods), instructors can easily emphasize one rule more than another to meet their students' needs. This approach also extends to modeling, applications, and graphing calculator use. The use of technology, which helps students visualize mathematical concepts, is flexible throughout the text, though technology use is not a requirement for students to benefit from this applications based book. The text contains numerous applications, including modeling of real-world data with functions and problem solving strategies. It is not necessary for an instructor to discuss any particular application; rather an instructor has the option to pick and choose from a wide variety of topics.
Changes to the Second Edition
Several important changes have been made to the Second Edition, which are a result of the many comments and suggestions received from instructors, students, and reviewers. The Second Edition is reorganized into eight chapters rather than six chapters. It has increased both the breadth and depth of the discussion for several topics, and added hundreds of new examples and exercises that involve mathematical skill building, applications, and modeling. More examples have been added for students to refer to when doing their homework assignments. Many of the real data applications from the First Edition have been updated to make the data more current and relevant to students. The new organization and seamless presentation allow students to easily understand each topic. The extensive exercise sets are carefully graded, covering a diverse assortment of topics with several levels of difficulty. The presentation and exercise sets provide instructors with the ability to enhance their courses in any of several different fashions. The chapter summaries have been expanded and presented in an easy-to-read grid format.
Chapter 1: Introduction to Functions and Graphs
This chapter has been streamlined from six sections to four. A basic introduction to problem solving is presented in Section 1.1 and the distance formula is now given in Section 1.2. Functions and their representation are introduced in Section 1.3, providing the background necessary for future chapters. The material on quadratic functions and transformations of graphs now appears in Chapter 3. Discussion of increasing and decreasing functions is delayed until Chapter 4.
Chapter 2: Linear Functions and Equations
Section 2.1 is a new section that
discusses linear functions and models in detail. Slope is discussed extensively
as a rate of change. Equations of lines are now presented before linear
equations and inequalities. There is an increased emphasis on problem solving,
which now includes several classical word problems. An extensive discussion of
piecewise linear functions, including absolute value equations and inequalities,
is provided in Section 2.5. Linear regression is introduced in Section 2.6.
Chapter 3: Quadratic Functions and Equations
This new chapter allows an instructor to introduce
quadratic functions and equations before covering the more involved nonlinear
functions and equations found in Chapter 4. Increased emphasis has been given to
problem solving. Quadratic equations and inequalities have been separated into
two sections. Transformations of graphs are introduced in Section 3.4, using
parabolas and other basic functions.
Chapter 4: Nonlinear Functions and Equations
In this chapter, the graphs, zeros, and factorization of
polynomials are discussed. Section 4.4 has been reorganized so that instructors
are able to cover complex numbers and complex solutions to quadratic equations
without covering the fundamental theorem of algebra. More opportunities for
modeling with nonlinear functions are provided. Section 4.6, Polynomial and
Rational Inequalities and Section 4.7, Power Functions and Radical Equations can
easily be omitted, but provide more opportunities for students to study other
nonlinear functions, equations, and inequalities.
Chapter 5: Exponential and Logarithmic Functions
In this chapter, operations on
functions and inverse functions are introduced in Sections 5.1 and 5.2. Section
5.3, Exponential Functions and Models, includes more discussion on the
differences between exponential growth and other types of growth. More
opportunities for modeling real data with exponential and logarithmic functions
are provided throughout the chapter. Logarithmic functions and properties of
logarithms are now separated into two sections to provide students with more
opportunities to increase skills and work with models. Section 5.7, Constructing
Exponential, Logarithmic, and Logistic Models, is a new section based on
nonlinear regression.
Chapter 6: Trigonometric Functions
This chapter provides a
comprehensive introduction to trigonometry, including both a right
triangle and a unit circle approach. Graphs of trigonometric functions and
inverse trigonometric functions are presented. Periodic data is modeled with
trigonometric functions.
Chapter 7: Trigonometric Identities and Equations
This chapter
includes additional practice with verifying identities, including applications
of identities and trigonometric equations. An introduction to solving inverse
trigonometric equations has been added. Presentations covering identities
and trigonometric equations are clear and understandable.
Chapter 8: Further Topics in Trigonometry
This chapter includes
material covering triangles, vectors, parametric equations, polar equations, and
complex numbers. New material covering parametric and polar equations has been
added to better prepare students for calculus, including additional examples and
exercises.
Chapter 9: Systems of Equations and Inequalities
This chapter now provides more examples and exercises for students to practice the skills needed to solve systems of linear and nonlinear equations. The material on linear programming has been increased.
Chapter 10: Conic Sections
This new chapter has expanded the material on conic sections from one section to three, providing a more comprehensive discussion with additional exercises. More practice solving nonlinear equations has also been included.
Chapter 11: Further Topics in Algebra
This chapter covers sequences, series, counting, probability, and the binomial theorem. Additional examples and exercises have been added throughout the chapter. The binomial theorem is presented in its own section. Additional material has been added to Section 8.5, Probability, which includes conditional probability and dependent events.
Chapter R: Basic Concepts from Algebra and Geometry
This new chapter provides a review of essential material from prerequisite courses, concentrating on skills used in algebra and geometry. Students are referred to this chapter by Algebra and Geometry Review Notes placed in the margins throughout the text. For example, students who need extra practice factoring trinomials are referred "just in time" to the proper section in Chapter R.
Appendix A: A Library of Functions
This appendix is new and summarizes many of the basic functions and families of functions.
Appendix B: The TI-83/83 Plus Graphing Calculators
This appendix has been expanded from the First Edition to provide more help with graphing calculators. It contains specific keystrokes for working selected examples from the text.
Features
Chapter and Section Introductions
Many college algebra
students have little or no understanding of what mathematics is about. Chapter and section introductions motivate and explain some of the reasons for studying mathematics. (See pages 1, 65, 157, 221, and 268.)
Applications and Models
Interesting, relevant applications are a major strength of
this textbook, and as a result, students become more effective problem solvers
and have a better understanding of how mathematics is used in the real world.
The applications are intuitive and not overly technical, so that they can be
introduced with a minimum amount of class time. Current data are used to create
meaningful mathematical models. A unique feature of this text is that the
applications and models are woven into both the discussions and the exercises.
It is easier for students to learn how to solve applications if they are
discussed within the text. (See pages 26, 165-166, 218, 334, and 349-351.)
An Index of Applications is included at the back of the text.
Sources
Since there are numerous applications throughout the text, genuine sources and a comprehensive bibliography are given. These sources reinforce for the student the practical applications of mathematics in real life. (See pages 104-106, and 387-390.)
Algebra and Geometry Review Notes
Throughout the text, there are Algebra and Geometry review notes located in the margins, which direct students "just in time" to Chapter R, where important topics in algebra and geometry are reviewed. Instructors can use this chapter for extra review or refer students to it as needed. This feature frees instructors from frequently reviewing material from intermediate algebra and geometry. (See pages 77 and 101.)
Graphing Calculator Help Notes
Throughout the text, there are Graphing Calculator Help Notes located in the margins, which direct students "just in time" to Appendix B: The TI-83/83 Plus Graphing Calculator. In this appendix, students are shown the necessary keystrokes to complete specific examples from the text. This feature frees instructors from teaching the specifics of the graphing calculator and gives students a convenient reference written specifically for this text. (See pages 7, 79, and 183.)
Making Connections
This
feature occurs throughout the text and helps students see how previous concepts covered are related to new concepts being presented. (See pages 30, 84, and 179.)
Putting It All Together
This helpful feature occurs at the end of each section to summarize techniques and reinforce the mathematical concepts presented in the section. It is given in an easy-to-follow grid format. (See pages 52-5, 319-320, and 383-384.)
Checking Basic Concepts
This
feature includes a small set of exercises provided after every two sections that can be used for review. These exercises require about 15 or 20 minutes to complete and can be used for collaborative learning exercises if time permits. (See pages 91, 188 and 311.)
Class Discussion
This feature is included in most sections and poses a question that can be used for either classroom discussion or homework. (See pages 32, 124, and 229.)
Extended and Discovery Exercises
Extended and Discovery Exercises occur at the end of selected sections and at the end of each chapter. These exercises are usually more complex, and often require discovery or extension of a topic presented in the chapter. They can be used for either collaborative learning or extra homework assignments. (See pages 172-173, 218-219, and 446-447.)
Exercise Sets
The exercise sets are the heart of any mathematics text, and this text includes a wide variety of instructive exercises. Each set contains exercises involving skill building, mathematical concepts, and applications. Graphical interpretation and tables of data are often used to extend students’ understanding of mathematical concepts. The exercise sets are carefully graded, and categorized according to topic, making it easier for an instructor to select an appropriate assignment. (See pages 169-173 and 346-352.)
Chapter Review Exercises
Chapter review exercises contain both skill-building exercises and applied exercises. They stress different techniques for solving problems, and provide students with the review necessary to successfully pass a chapter test. (See pages 60-62 and 328-331.)
Chapter Summaries
Chapter summaries are expanded and presented in an easy-to-read grid format. (See pages 213-215 and 324-327.)
