Precalculus Through Modeling and Visualization

by Rockswold, Gary K.

• ISBN13: 9780321082015
• ISBN10: 032108201X
• Edition: 2nd
• Format: Hardcover
• Copyright: 2002-01-01
• Publisher: Addison Wesley

Summary

This book offers an innovative approach by focusing on learning through modeling and visualization with an integrated use of the graphing calculator. When introducing mathematical ideas, this text moves from the concrete to the abstract. The importance of mathematics is thus demonstrated to students and the material is presented in an accessible manner. The text consistently integrates mathematical concepts with real applications in order to enhance student intuition and understanding. Symbolic (algebraic), graphical, numerical, and verbal skills are continually demonstrated and reinforced throughout.

Table of Contents

Preface	xiii
Introduction to Functions and Graphs	1	(63)
Numbers, Data, and Problem Solving	2	(11)
Sets of Numbers
Scientific Notation
Problem Solving
Visualization of Data	13	(12)
One-Variable Data
Two-Variable Data
The Distance Formula
Visualizing Data with a Graphing Calculator
Checking Basic Concepts for Sections 1.1 and 1.2	25	(1)
Functions and Their Representations	25	(16)
Basic Concepts
Representations of Functions
Formal Definition of a Function
Graphing Calculators and Functions
Identifying Functions
Types of Functions and Their Rates of Change	41	(23)
Constant Functions Linear Functions
Slope as a Rate of Change
Nonlinear Functions
Average Rate of Change
Checking Basic Concepts for Sections 1.3 and 1.4	57	(1)
Summary	58	(2)
Review Exercises	60	(2)
Extended and Discovery Exercises	62	(2)
Linear Functions and Equations	64	(93)
Linear Functions and Models	65	(9)
Exact and Approximate Models
Graphs of Linear Functions
Modeling with Linear Functions
Equations of Lines	74	(17)
Forms for Equations of Lines
Determining Intercepts
Horizontal, Vertical, Parallel, and Perpendicular Lines
Direct Variation
Checking Basic Concepts for Sections 2.1 and 2.2	91	(1)
Linear Equations	91	(16)
Equations
The Intersection-of-Graphs Method
Solving Linear Equations Symbolically
Problem-Solving Strategies
Linear Inequalities	107	(15)
Inequalities
Techniques for Solving Inequalities
Compound Inequalities
Interval Notation
Checking Basic Concepts for Sections 2.3 and 2.4	122	(1)
Piecewise-Defined Linear Functions	122	(16)
Evaluating and Graphing Piecewise-Defined Functions
The Greatest Integer Function
The Absolute Value Function
Equations and Inequalities
Involving Absolute Values
Linear Approximation	138	(19)
Midpoint Formula
Interpolation and Extrapolation
Linear Regression
Checking Basic Concepts for Sections 2.5 and 2.6	149	(1)
Summary	149	(3)
Review Exercises	152	(3)
Extended and Discovery Exercises	155	(2)
Quadratic Functions and Equations	157	(63)
Quadratic Functions and Models	158	(15)
Basic Concepts
Vertex Form
Models and Applications
Quadratic Regression
Quadratic Equations and Problem Solving	173	(15)
Basic Concepts
Solving Quadratic Equations
Problem Solving
Checking Basic Concepts for Sections 3.1 and 3.2	188	(1)
Quadratic Inequalities	188	(8)
Graphical Solutions
Symbolic and Numerical Solutions
Transformations of Graphs	196	(24)
Vertical and Horizontal Translations
Stretching and Shrinking
Reflections of Graphs
Dynamically Displayed Data: An Application
Checking Basic Concepts for Sections 3.3 and 3.4	213	(1)
Summary	213	(3)
Review Exercises	216	(2)
Extended and Discovery Exercises	218	(2)
Nonlinear Functions and Equations	220	(113)
Nonlinear Functions and Their Graphs	221	(15)
Polynomial Functions
Increasing and Decreasing Functions
Extrema of Nonlinear Functions
Symmetry
Polynomial Functions and Models	236	(15)
Graphs of Polynomial Functions
Polynomial Regression
Piecewise-Defined Polynomial Functions
Checking Basic Concepts for Sections 4.1 and 4.2	251	(1)
Real Zeros of Polynomial Functions	251	(17)
Division of Polynomials
Factoring Polynomials
Graphs and Multiple Zeros
Rational Zeros
Polynomial Equations
The Fundamental Theorem of Algebra	268	(14)
Complex Numbers
Quadratic Equations with Complex Solutions
Fundamental Theorem of Algebra
Polynomial Equations with Complex Solutions
Checking Basic Concepts for Sections 4.3 and 4.4	282	(1)
Rational Functions and Models	282	(20)
Rational Functions
Vertical Asymptotes
Horizontal Asymptotes
Identifying Asymptotes
Rational Equations
Variation
Polynomial and Rational Inequalities	302	(10)
Polynomial Inequalities
Rational Inequalities
Checking Basic Concepts for Sections 4.5 and 4.6	311	(1)
Power Functions and Radical Equations	312	(21)
Rational Exponents and Radical Notation
Power Functions and Models
Equations Involving Radicals
Power Regression
Checking Basic Concepts for Section 4.7	323	(1)
Summary	324	(4)
Review Exercises	328	(3)
Extended and Discovery Exercises	331	(2)
Exponential and Logarithmic Functions	333	(115)
Combining Functions	334	(18)
Arithmetic Operations on Functions
Composition of Functions
Inverse Functions and Their Representations	352	(18)
Inverse Operations
One-to-One Functions
Symbolic Representations of Inverse Functions
Other Representations of Inverse Functions
Checking Basic Concepts for Sections 5.1 and 5.2	370	(1)
Exponential Functions and Models	370	(20)
Linear and Exponential Growth
Exponential Models
Compound Interest
The Natural Exponential Function
Logarithmic Functions and Models	390	(17)
The Common Logarithmic Function
Solving Equations
Logarithms with Other Bases
General Logarithmic Equations
Checking Basic Concepts for Sections 5.3 and 5.4	407	(1)
Properties of Logarithms	407	(8)
Basic Properties of Logarithms
Change of Base Formula
Exponential and Logarithmic Equations	415	(15)
Exponential Equations
Logarithmic Equations
Checking Basic Concepts for Sections 5.5 and 5.6	429	(1)
Constructing Nonlinear Models	430	(18)
Exponential Model
Logarithmic Model
Logistic Model
Checking Basic Concepts for Section 5.7	438	(1)
Summary	438	(4)
Review Exercises	442	(4)
Extended and Discovery Exercises	446	(2)
Trigonometric Functions	448	(103)
Angles and Their Measure	449	(15)
Angles
Degree Measure
Radian Measure
Arc Length
Area of a Sector
Right Triangle Trigonometry	464	(14)
Basic Concepts of Trigonometric Functions
Applications of Right Triangle Trigonometry
Complementary Angles and Cofunctions
Checking Basic Concepts for Sections 6.1 and 6.2	478	(1)
The Sine and Cosine Functions and Their Graphs	478	(15)
Definitions
The Unit Circle
Representations of the Sine and the Cosine Functions
Modeling with the Sine and Cosine Functions
Other Trigonometric Functions and Their Graphs	493	(14)
Definitions and Basic Identities
Representations of Other Trigonometric Functions
Modeling with Trigonometric Functions
Checking Basic Concepts for Sections 6.3 and 6.4	507	(1)
Modeling with Trigonometric Functions	507	(18)
Transformations of Trigonometric Graphs
Models Involving Trigonometric Functions
Simple Harmonic Motion
Inverse Trigonometric Functions	525	(26)
Review of Inverses
The Inverse Sine Function
The Inverse Cosine Function
The Inverse Tangent Function
Solving Triangles and Equations
Checking Basic Concepts for Sections 6.5 and 6.6	542	(1)
Summary	542	(3)
Review Exercises	545	(4)
Extended and Discovery Exercises	549	(2)
Trigonometric Identities and Equations	551	(77)
Fundamental Identities	552	(13)
Reciprocal and Quotient Identities
Pythagorean Identities
Negative-Angle Identities
Verifying Identities	565	(8)
Simplifying Trigonometric Expressions
Verification of Identities
Checking Basic Concepts for Sections 7.1 and 7.2	573	(1)
Trigonometric Equations	573	(16)
Reference Angles
Solving Trigonometric Equations
Solving Inverse Trigonometric Equations
Sum and Difference Identities	589	(12)
Sum and Difference Identities for Cosine
Other Sum and Difference Identities
Checking Basic Concepts for Sections 7.3 and 7.4	600	(1)
Multiple-Angle Identities	601	(27)
Double-Angle Identities
Half-Angle Formulas
Solving Equations
Product-to-Sum and Sum-to-Product Identities
Checking Basic Concepts for Section 7.5	619	(1)
Summary	619	(4)
Review Exercises	623	(3)
Extended and Discovery Exercises	626	(2)
Further Topics in Trigonometry	628	(82)
Law of Sines	629	(11)
Oblique Triangles
Solving Triangles
The Ambiguous Case
Law of Cosines	640	(10)
Derivation of the Law of Cosines
Solving Triangles
Area Formulas
Checking Basic Concepts for Sections 8.1 and 8.2	649	(1)
Vectors	650	(17)
Basic Concepts
Operations on Vectors
The Dot Product
Work
Parametric Equations	667	(10)
Basic Concepts
Application of Parametric Equations
Checking Basic Concepts for Sections 8.3 and 8.4	677	(1)
Polar Equations	677	(13)
The Polar Coordinate System
Graphs of Polar Equations
Solving Polar Equations
Trigonometric Form and Roots of Complex Numbers	690	(20)
Trigonometric Form
Products and Quotients of Complex Numbers De Moivre's Theorem
Roots of Complex Numbers
Checking Basic Concepts for Sections 8.5 and 8.6	702	(1)
Summary	702	(3)
Review Exercises	705	(3)
Extended and Discovery Exercises	708	(2)
Systems of Equations and Inequalities	710	(98)
Functions and Equations of Two Variables	711	(15)
Functions of Two Variables
Systems of Equations
The Method of Substitution
Graphical and Numerical Methods
Joint Variation
Linear Systems of Equations and Inequalities in Two Variables	726	(18)
Types of Linear Systems in Two Variables
The Elimination Method Systems of Inequalities
Linear Programming
Checking Basic Concepts for Sections 9.1 and 9.2	744	(1)
Solutions of Linear Systems Using Matrices	744	(16)
Representing Systems of Linear Equations with Matrices
Row-Echelon Form
Gaussian Elimination
Solving Systems of Linear Equations with Technology
Properties and Applications of Matrices	760	(15)
Matrix Notation
Sums, Differences, and Scalar Multiples of Matrices
Matrix Products
Technology and Matrices
Checking Basic Concepts for Sections 9.3 and 9.4	775	(1)
Inverses of Matrices	775	(13)
Matrix Inverses
Representing Linear Systems with Matrix Equations
Solving Linear Systems with Inverses
Finding Inverses Symbolically
Determinants	788	(20)
Definition and Calculation of Determinants
Area of Regions
Cramer's Rule
Limitations on the Method of Cofactors and Cramer's Rule
Checking Basic Concepts for Sections 9.5 and 9.6	798	(1)
Summary	798	(4)
Review Exercises	802	(3)
Extended and Discovery Exercises	805	(3)
Conic Sections	808	(44)
Parabolas	809	(12)
Equations and Graphs of Parabolas
Reflective Property of Parabolas
Translations of Parabolas
Ellipses	821	(16)
Equations and Graphs of Ellipses
Translations of Ellipses
Reflective Property of Ellipses
Circles
Solving Systems of Equations and Inequalities
Checking Basic Concepts for Sections 10.1 and 10.2	837	(1)
Hyperbolas	837	(15)
Equations and Graphs of Hyperbolas
Translations of Hyperbolas
Reflective Property of Hyperbolas
Checking Basic Concepts for Sections 10.3	847	(1)
Summary	847	(3)
Review Exercises	850	(1)
Extended and Discovery Exercises	851	(1)
Further Topics in Algebra	852	(1)
Sequences	853	(14)
Sequences and Functions
Representations of Sequences
Arithmetic Sequences
Geometric Sequences
Series	867	(15)
Basic Concepts
Arithmetic Series
Geometric Series
Summation Notation
Checking Basic Concepts for Sections 11.1 and 11.2	882	(1)
Counting	882	(11)
Fundamental Counting Principle
Permutations
Combinations
The Binomial Theorem	893	(5)
Derivation of the Binomial Theorem
Pascal's Triangle
Checking Basic Concepts for Sections 11.3 and 11.4	898	(1)
Probability	898	(14)
Definition of Probability
Compound Events
Independent and Dependent Events
Checking Basic Concepts for Section 11.5	912	(1)
Summary	912	(4)
Review Exercises	916	(1)
Extended and Discovery Exercises	917
Reference: Basic Concepts from Algebra and Geometry	1	(1)
Formulas from Geometry	1	(9)
Geometric Shapes in a Plane
The Pythagorean Theorem
Three-Dimensional Objects
Similar Triangles
A Summary of Geometric Formulas
Circles	10	(4)
Equations and Graphs of Circles
Finding the Center and Radius of a Circle
Integer Exponents	14	(6)
Bases and Positive Exponents
Zero and Negative Exponents
Product, Quotient, and Power Rules
Polynomial Expressions	20	(7)
Addition and Subtraction of Monomials
Addition and Subtraction of Polynomials
Distributive Properties
Multiplying Polynomials
Some Special Products
Factoring Polynomials	27	(7)
Common Factors
Factoring by Grouping
Factoring Trinomials
Difference of Two Squares
Perfect Square Trinomials
Sum and Difference of Two Cubes
Rational Expressions	34
Multiplication and Division of Rational Expressions
Common Denominators
Addition and Subtraction of Rational Expressions
Clearing Fractions
Complex Fractions
Appendix A A Library of Functions	1	(4)
Appendix B The TI-83/83 Plus Graphing Calculators	5
Bibliography	1	(1)
Answers to Selected Exercises	1	(1)
Index of Applications	1	(6)
Index	7