Student Solutions Manual for College Algebra and Trigonometry and Precalculus

Description

• Over 20% of exercises are updated, and a substantial amount of brand-new exercises have been added.

• An improved balance of exercises provides a smoother transition from the less-challenging to the more-challenging exercises. Well-crafted Exercises are offered in a quantity, quality, and variety that meets the needs of all students. They are clearly labeled for instructors and students understanding of exercise expectations.

• Getting Ready for the Next Section exercises end each exercise set with problems that provide a review of concepts and skills that will be used in the following section.

• Graph and Data-Related exercises—We have introduced exercises throughout the text that demonstrate how to extract information about real-world situations from a graphical representation of that situation, as well as how to recover algebraic or trigonometric formulations of a graph by using key characteristics of that graph.

• Modeling Exercises—A section on building linear, exponential, logarithmic, and power models from data was added in Chapter 4 that contains new exercises using each type of model.

• Concepts and Vocabulary exercises begin each exercise set with problems that assess the student’s grasp of the definitions and ideas introduced in that section. These true/false and fill-in-the-blank exercises help to rapidly identify gaps in comprehension of the material in that section.

• Exercises preparing students for material in the next section—Each exercise section ends with a set of exercises titled Getting Ready for the Next Section that provides a review of the concepts and skills that will be used in the following section.

Table of Contents

1. BASIC CONCEPTS OF ALGEBRA
   • P.1 The Real Numbers and Their Properties
   • P.2 Integer Exponents and Scientific Notation
   • P.3 Polynomials
   • P.4 Factoring Polynomials
   • P.5 Rational Expressions
   • P.6 Rational Exponents and Radicals
   • Chapter P Review and Tests
   • Review Exercises
   • Practice Test
2. EQUATIONS AND INEQUALITIES
   • 1.1 Linear Equations in One Variable
   • 1.2 Applications of Linear Equations: Modeling
   • 1.3 Quadratic Equations
   • 1.4 Complex Numbers: Quadratic Equations with Complex Solutions
   • 1.5 Solving Other Types of Equations
   • 1.6 Inequalities
   • 1.7 Equations and Inequalities Involving Absolute Value
   • Chapter 1 Review and Tests
   • Review Exercises
   • Practice Test A
   • Practice Test B
3. GRAPHS AND FUNCTIONS
   • 2.1 The Coordinate Plane
   • 2.2 Graphs of Equations
   • 2.3 Lines
   • 2.4 Functions
   • 2.5 Properties of Functions
   • 2.6 A Library of Functions
   • 2.7 Transformations of Functions
   • 2.8 Combining Functions; Composite Functions
   • 2.9 Inverse Functions
   • Chapter 2 Review and Tests
   • Review Exercises
   • Practice Test A
   • Practice Test B
   • Cumulative Review Exercises Chapters P—2
4. POLYNOMIAL AND RATIONAL FUNCTIONS
   • 3.1 Quadratic Functions
   • 3.2 Polynomial Functions
   • 3.3 Dividing Polynomials
   • 3.4 The Real Zeros of a Polynomial Function
   • 3.5 The Complex Zeros of a Polynomial Function
   • 3.6 Rational Functions
   • 3.7 Variation
   • Chapter 3 Review and Tests
   • Review Exercises
   • Practice Test A
   • Practice Test B
   • Cumulative Review Exercises Chapters P—3
5. EXPONENTIAL AND LOGARITHMIC FUNCTIONS
   • 4.1 Exponential Functions
   • 4.2 Logarithmic Functions
   • 4.3 Rules of Logarithms
   • 4.4 Exponential and Logarithmic Equations and Inequalities
   • 4.5 Logarithmic Scales; Modeling
   • Chapter 4 Review and Tests
   • Review Exercises
   • Practice Test A
   • Practice Test B
   • Cumulative Review Exercises Chapters P—4
6. TRIGONOMETRIC FUNCTIONS
   • 5.1 Angles and Their Measure
   • 5.2 Right-Triangle Trigonometry
   • 5.3 Trigonometric Functions of Any Angle; The Unit Circle
   • 5.4 Graphs of the Sine and Cosine Functions
   • 5.5 Graphs of the Other Trigonometric Functions
   • 5.6 Inverse Trigonometric Functions
   • Chapter 5 Review and Tests
   • Review Exercises
   • Practice Test A
   • Practice Test B
   • Cumulative Review Exercises Chapters P—5
7. TRIGONOMETRIC IDENTITIES AND EQUATIONS
   • 6.1 Trigonometric Identities
   • 6.2 Sum and Difference Formulas
   • 6.3 Double-Angle and Half-Angle Formulas
   • 6.4 Product-to-Sum and Sum-to-Product Formulas
   • 6.5 Trigonometric Equations I
   • 6.6 Trigonometric Equations II
   • Chapter 6 Review and Tests
   • Review Exercises
   • Practice Test A
   • Practice Test B
   • Cumulative Review Exercises Chapters P—6
8. APPLICATIONS OF TRIGONOMETRIC FUNCTIONS
   • 7.1 The Law of Sines
   • 7.2 The Law of Cosines
   • 7.3 Areas of Polygons Using Trigonometry
   • 7.4 Vectors
   • 7.5 The Dot Product
   • 7.6 Polar Coordinates
   • 7.7 Polar Form of Complex Numbers; DeMoivre’s Theorem
   • Chapter 7 Review and Tests
   • Review Exercises
   • Practice Test A
   • Practice Test B
   • Cumulative Review Exercises Chapters P—7
9. SYSTEMS OF EQUATIONS AND INEQUALITIES
   • 8.1 Systems of Linear Equations in Two Variables
   • 8.2 Systems of Linear Equations in Three Variables
   • 8.3 Partial-Fraction Decomposition
   • 8.4 Systems of Nonlinear Equations
   • 8.5 Systems of Inequalities
   • 8.6 Linear Programming
   • Chapter 8 Review and Tests
   • Review Exercises
   • Practice Test A
   • Practice Test B
   • Cumulative Review Exercises Chapters P—8
10. MATRICES AND DETERMINANTS
    • 9.1 Matrices and Systems of Equations
    • 9.2 Matrix Algebra
    • 9.3 The Matrix Inverse
    • 9.4 Determinants and Cramer’s Rule
    • Chapter 9 Review and Tests
    • Review Exercises
    • Practice Test A
    • Practice Test B
    • Cumulative Review Exercises Chapters P—9
11. CONIC SECTIONS
    • 10.1 Conic Sections: Overview
    • 10.2 The Parabola
    • 10.3 The Ellipse
    • 10.4 The Hyperbola
    • Chapter 10 Review and Tests
    • Review Exercises
    • Practice Test A
    • Practice Test B
    • Cumulative Review Exercises Chapters P—10
12. FURTHER TOPICS IN ALGEBRA
    • 11.1 Sequences and Series
    • 11.2 Arithmetic Sequences; Partial Sums
    • 11.3 Geometric Sequences and Series
    • 11.4 Mathematical Induction
    • 11.5 The Binomial Theorem
    • 11.6 Counting Principles
    • 11.7 Probability
    • Chapter 11 Review and Tests
    • Review Exercises
    • Practice Test A
    • Practice Test B
    • Cumulative Review Exercises Chapters P—11

Answers to Selected Exercises

Credits

Index

• Chapter openers: Each chapter opener includes a description of applications (one of them illustrated) relevant to the content of the chapter and the list of topics that will be covered. In one page, students see what they are going to learn and why they are learning it.

• Section opener with application:  Each section opens with a list of prerequisite topics, complete with section and page references, which students can review prior to starting the section.

• The Objectives of the section are also clearly stated and numbered, and then referenced again in the margin of the lesson at the point where the objective’s topic is taught.

• An Application then follows, containing  a  motivating  anecdote  or  an interesting problem.

• An example later in the section relating to this application and identified by the same icon is then solved using the mathematics covered in the section. These applications utilize material from a variety of fields: the physical and biological sciences (including health sciences), economics, art and architecture, history, and more.

• Definitions, Theorems, Properties, and Rules are all boxed and titled for emphasis and ease of reference.

• Work it out: Ratti/McWaters/Skryzpek encourage students to practice the material frequently, and give them ample opportunities to master the material by solving problems and applying their understanding.

• Procedure in Action is a special feature which introduces procedure steps within the context of a worked-out example. Important multistep procedures, such as the steps for finding the inverse of a one-to-one function, are presented in a two-column format.

• Examples include a wide range of computational, conceptual, and modern applied problems carefully selected to build confidence, competency, and understanding. Every example has a title indicating its purpose and presents a detailed solution containing annotated steps.

• All examples are followed by a Practice Problem for students to try so that they can check their understanding of the concept covered.

• UPDATED! Over 20% of exercises are updated, and over 500 brand-new exercises have been added.  New exercises primarily consist of applications that connect with students everyday experiences and enhance students conceptual understanding of graphing.

• REVISED! An improved balance of exercises provides a smoother transition from the less-challenging to the more-challenging exercises. Well-crafted Exercises are offered in a quantity, quality, and variety that meets the needs of all students. They are clearly labeled for instructors and students understanding of exercise expectations.

The Basics

• NEW! Concepts and Vocabulary exercises begin each exercise set with problems that assess the student’s grasp of the definitions and ideas introduced in that section. These true/false and fill-in-the-blank exercises help to rapidly identify gaps in comprehension of the material in that section.

• Building Skills exercises develop fundamental skills; each odd-numbered exercise is closely paired with its consecutive even-numbered exercise.

Applications

• Applying the Concepts features use the section’s material to solve real-world problems; all are titled and relevant to the topics of the section.

Extensions

• Beyond the Basics exercises provide more challenging problems that give students an opportunity to reach beyond the material covered in the section; these are generally more theoretical in nature and are suitable for honors students, special assignments, or extra credit.

• NEW! Graph and Data-Related Exercises. These demonstrate how to extract information about real-world situations from a graphical representation of that situation, as well as how to recover algebraic or trigonometric formulations of a graph by using key characteristics of that graph.

• Critical Thinking/Discussion/Writing exercises, placed appropriately, are designed to develop students’ higher-level thinking skills. Calculator problems, identified by an icon, are included where needed.

• NEW! Getting Ready for the Next Section exercises end each exercise set with problems that provide a review of concepts and skills that will be used in the following section.

• NEW! Modeling Exercises. A section on building linear, exponential, logarithmic, and power models from data in Chapter 4 contains new exercises using each type of model.

• Help along the way: These integrated study aids give students hints and tips at strategic places in the text, addressing some of the most frequent issues and questions that occur in office hours.

• Summary of Main Facts boxes summarize information related to equations and their graphs, such as those of the conic sections.

• Side Notes provide hints for handling newly introduced concepts.

• Recall notes remind students of a key idea learned earlier in the text that will help them work through a current problem.

• Warnings appear as appropriate throughout the text to apprise students of common errors and pitfalls that can trip them up in their thinking or calculations.

• Technology Connections give students tips on using calculators to solve problems, check answers, and reinforce concepts. Note that the use of graphing calculators is optional in this text.

• Do You Know? features provide students with additional interesting information on topics to keep them engaged in the mathematics presented.

• Historical Notes give students information on key people or ideas in the history and development of mathematics.

• Preparation and review: End-of-chapter material includes all of the following items to help students prepare for exams and make the most of their study time.  

• An Updated Summary starts the end-of-chapter material, featuring an extensive review of the definitions, concepts, and formulas covered in that chapter with corresponding examples. This Review provides a description and examples of key topics indicating where the material occurs in the text, and encourages students to reread sections rather than memorize definitions out of context.

• Review Exercises provide students with an opportunity to practice what they have learned in the chapter.

• Two full Practice Tests—Practice Test A in the usual open-ended format and Practice Test B, covering the same topics, in a multiple-choice format.

• Cumulative Review Exercises appear at the end of every chapter, starting with Chapter 2, to remind students that mathematics is not modular and that what is learned in the first part of the book will be useful in later parts of the book and on the final examination.

J.S. Ratti has been teaching mathematics at all levels for over 35 years. He is currently a full professor and past chair of mathematics at the University of South Florida. Professor Ratti is the author of numerous research papers in analysis, graph theory, and probability. He has received several awards–including a USF Research Council Grant, USF Teaching Incentive Program (TIP) Award, USF Outstanding Undergraduate Teaching Award, and Academy of Applied Sciences grants–and is the coauthor of a successful finite mathematics textbook. He enjoys both college and professional football as well as traveling.

Marcus McWaters is currently an Associate Professor at the University of South Florida (USF). He is a former Chair of the Department of Mathematics and Statistics at USF. Since receiving his PhD in mathematics from the University of Florida, he has taught all levels of undergraduate and graduate courses, with class sizes ranging from 3 to 250. As Chair, he successfully structured a course delivery system for lower-level courses that improved the low retention rate in those courses at USF. He is also a founding member of the USF Center for Digital and Computational Video. When not involved with mathematics or administrative activity, he enjoys traveling with his wife and two daughters, theater, water¿skiing, and racquetball.

Leslaw Skrzypek is currently the Chair of the Department of Mathematics and Statistics at the University of South Florida. His research is in the area of Banach Spaces and Approximation Theory. He is the recipient of a Fulbright Award and a NATO Advanced Grant research award, and is a founding director of the USF Center for Complex Data Systems. Throughout his career, Professor Skrzypek has enjoyed teaching all levels of courses, from remedial to graduate real analysis. Over the years he also has been involved in training students for the Mathematical Olympiads. He enjoys nature, listening to music, and spending time with his family.