Essentials of precalculus

Description

Front Cover Title Page Copyright Contents Preface 1 FUNCTIONS AND GRAPHS 1.1 ONS AND I NEQUALI TI ES Equations and Inequalities ABSOLUTE VALUE AND THE REAL NUMBERS ABSOLUTE VALUE AND DISTANCE LINEAR AND QUADRATIC EQUATIONS LINEAR AND QUADRATIC INEQUALITIES ABSOLUTE VALUE INEQUALITIES 1.2 A Two-Dimensional Coordinate System and Graphs CARTESIAN COORDINATE SYSTEMS THE DISTANCE AND MIDPOINT FORMULAS GRAPH OF AN EQUATION INTERCEPTS CIRCLES,THEIR EQUATIONS, AND THEIR GRAPHS 1.3 Introduction to Functions RELATIONS FUNCTIONS FUNCTIONAL NOTATION IDENTIFYING FUNCTIONS GRAPHS OF FUNCTIONS THE GREATEST INTEGER FUNCTION (FLOOR FUNCTION) APPLICATIONS 1.4 Linear Functions SLOPES OF LINES FIND THE EQUATION OF A LINE APPLICATIONS PARALLEL AND PERPENDICULAR LINES 1.5 Quadratic Functions VERTEX OF A PARABOLA MAXIMUM AND MINIMUM OF A QUADRATIC FUNCTION APPLICATIONS 1.6 Properties of Graphs SYMMETRY EVEN AND ODD FUNCTIONS TRANSLATIONS OF GRAPHS REFLECTIONS OF GRAPHS COMPRESSING AND STRETCHING OF GRAPHS 1.7 The Algebra of Functions OPERATIONS ON FUNCTIONS THE DIFFERENCE QUOTIENT COMPOSITION OF FUNCTIONS EXPLORING CONCEPTS WITH TECHNOLOGY: Graphing Piecewise Functions with a Graphing Calculator Chapter 1 Summary Chapter 1 True/False Exercises Chapter 1 Review Exercises Chapter 1 Test 2 POLYNOMIAL AND RATIONAL FUNCTIONS 2.1 Complex Numbers INTRODUCTION TO COMPLEX NUMBERS ADDITION AND SUBTRACTION OF COMPLEX NUMBERS MULTIPLICATION OF COMPLEX NUMBERS DIVISION OF COMPLEX NUMBERS POWERS OF i SOLVE QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS 2.2 The Remainder Theorem and the Factor Theorem DIVISION OF POLYNOMIALS THE REMAINDER THEOREM THE FACTOR THEOREM REDUCED POLYNOMIALS 2.3 Polynomial Functions of Higher Degree FAR-LEFT AND FAR-RIGHT BEHAVIOR MAXIMUM AND MINIMUM VALUES REAL ZEROS OF A POLYNOMIAL FUNCTION EVEN AND ODD POWERS OF (X – C) THEOREM A PROCEDURE FOR GRAPHING POLYNOMIAL FUNCTIONS 2.4 Zeros of Polynomial Functions MULTIPLE ZEROS OF A POLYNOMIAL FUNCTION THE RATIONAL ZERO THEOREM UPPER AND LOWER BOUNDS FOR REAL ZEROS DESCARTES’ RULE OF SIGNS ZEROS OF A POLYNOMIAL FUNCTION APPLICATIONS OF POLYNOMIAL FUNCTIONS 2.5 The Fundamental Theorem of Algebra THE FUNDAMENTAL THEOREM OF ALGEBRA THE NUMBER OF ZEROS OF A POLYNOMIAL FUNCTION THE CONJUGATE PAIR THEOREM FIND A POLYNOMIAL FUNCTION WITH GIVEN ZEROS 2.6 Graphs of Rational Functions and Their Applications VERTICAL AND HORIZONTAL ASYMPTOTES A SIGN PROPERTY OF RATIONAL FUNCTIONS A GENERAL GRAPHING PROCEDURE SLANT ASYMPTOTES GRAPH RATIONAL FUNCTIONS THAT HAVE A COMMON FACTOR APPLICATIONS OF RATIONAL FUNCTIONS EXPLORING CONCEPTS WITH TECHNOLOGY: Finding Zeros of a Polynomial Using Mathematica Chapter 2 Summary Chapter 2 True/False Exercises Chapter 2 Review Exercises Chapter 2 Test Cumulative Review Exercises 3 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 3.1 Inverse Functions INTRODUCTION TO INVERSE FUNCTIONS GRAPHS OF INVERSE FUNCTIONS COMPOSITION OF A FUNCTION AND ITS INVERSE FIND AN INVERSE FUNCTION APPLICATION 3.2 Exponential Functions and Their Applications EXPONENTIAL FUNCTIONS GRAPHS OF EXPONENTIAL FUNCTIONS THE NATURAL EXPONENTIAL FUNCTION APPLICATIONS OF EXPONENTIAL FUNCTIONS 3.3 Logarithmic Functions and Their Applications LOGARITHMIC FUNCTIONS GRAPHS OF LOGARITHMIC FUNCTIONS DOMAINS OF LOGARITHMIC FUNCTIONS COMMON AND NATURAL LOGARITHMS APPLICATIONS OF LOGARITHMIC FUNCTIONS 3.4 Logarithms and Logarithmic Scales PROPERTIES OF LOGARITHMS CHANGE-OF-BASE FORMULA LOGARITHMIC SCALES 3.5 Exponential and Logarithmic Equations SOLVE EXPONENTIAL EQUATIONS SOLVE LOGARITHMIC EQUATIONS APPLICATION 3.6 Exponential Growth and Decay COMPOUND INTEREST EXPONENTIAL GROWTH EXPONENTIAL DECAY CARBON DATING EXPLORING CONCEPTS WITH TECHNOLOGY: Using a Semilog Graph to Model Exponential Decay Chapter 3 Summary Chapter 3 True/False Exercises Chapter 3 Review Exercises Chapter 3 Test Cumulative Review Exercises 4 TRIGONOMETRIC FUNCTIONS 4.1 Angles and Arcs DEGREE MEASURE CLASSIFICATION OF ANGLES CONVERSION BETWEEN UNITS RADIAN MEASURE ARCS AND ARC LENGTH LINEAR AND ANGULAR SPEED 4.2 Trigonometric Functions of Real Numbers THEWRAPPING FUNCTION TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS PROPERTIES OF TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS TRIGONOMETRIC IDENTITIES AN APPLICATION INVOLVING A TRIGONOMETRIC FUNCTION 4.3 Graphs of the Sine and Cosine Functions THE GRAPH OF THE SINE FUNCTION THE GRAPH OF THE COSINE FUNCTION 4.4 Graphs of the Other Trigonometric Functions THE GRAPH OF THE TANGENT FUNCTION THE GRAPH OF THE COTANGENT FUNCTION THE GRAPH OF THE COSECANT FUNCTION THE GRAPH OF THE SECANT FUNCTION 4.5 Graphing Techniques TRANSLATION OF TRIGONOMETRIC FUNCTIONS ADDITION OF ORDINATES DAMPING FACTOR TOPICS FOR DISCUSSION EXPLORING CONCEPTS WITH TECHNOLOGY: Sinusoidal Families Chapter 4 Summary Chapter 4 True/False Exercises Chapter 4 Review Exercises Chapter 4 Test Cumulative Review Exercises 5 APPLICATIONS OF TRIGONOMETRY AND TRIGONOMETRIC IDENTITIES 5.1 Trigonometric Functions of Angles TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES APPLICATIONS INVOLVING RIGHT TRIANGLES TRIGONOMETRIC FUNCTIONS OF ANY ANGLE TRIGONOMETRIC FUNCTIONS OF QUADRANTAL ANGLES SIGNS OF TRIGONOMETRIC FUNCTIONS THE REFERENCE ANGLE 5.2 Verification of Trigonometric Identities VERIFICATION OF TRIGONOMETRIC IDENTITIES IDENTITIES THAT INVOLVE (α ± β) COFUNCTIONS ADDITIONAL SUM AND DIFFERENCE IDENTITIES 5.3 More on Trigonometric Identities DOUBLE-ANGLE IDENTITIES HALF-ANGLE IDENTITIES THE PRODUCT-TO-SUM IDENTITIES THE SUM-TO-PRODUCT IDENTITIES FUNCTIONS OF THE FORM f(x) = a sin x + b cos x 5.4 Inverse Trigonometric Functions INVERSE TRIGONOMETRIC FUNCTIONS COMPOSITION OF TRIGONOMETRIC FUNCTIONS AND THEIR INVERSES GRAPHS OF INVERSE TRIGONOMETRIC FUNCTIONS 5.5 Trigonometric Equations SOLVE TRIGONOMETRIC EQUATIONS AN APPLICATION INVOLVING A TRIGONOMETRIC EQUATION 5.6 The Law of Sines and the Law of Cosines THE LAW OF SINES THE AMBIGUOUS CASE (SSA) APPLICATIONS OF THE LAW OF SINES THE LAW OF COSINES AN APPLICATION OF THE LAW OF COSINES 5.7 Vectors VECTORS UNIT VECTORS APPLICATIONS OF VECTORS DOT PRODUCT SCALAR PROJECTION PARALLEL AND PERPENDICULAR VECTORS WORK:AN APPLICATION OF THE DOT PRODUCT EXPLORING CONCEPTS WITH TECHNOLOGY: Approximate an Inverse Trigonometric Function with Polynomials Chapter 5 Summary Chapter 5 True/False Exercises Chapter 5 Review Exercises Chapter 5 Test Cumulative Review Exercises 6 ADDITIONAL TOPICS IN MATHEMATICS 6.1 Conic Sections PARABOLAS PARABOLAS WITH VERTEX AT (h, k) ELLIPSES HYPERBOLAS APPLICATIONS 6.2 Polar Coordinates THE POLAR COORDINATE SYSTEM GRAPHS OF EQUATIONS IN A POLAR COORDINATE SYSTEM POLAR EQUATIONS OF THE CONICS GRAPH A CONIC GIVEN IN POLAR FORM TRANSFORMATIONS BETWEEN RECTANGULAR AND POLAR COORDINATES WRITE POLAR COORDINATE EQUATIONS AS RECTANGULAR EQUATIONS AND RECTANGULAR COORDINATE EQUATIONS AS POLAR EQUATIONS 6.3 Parametric Equations PARAMETRIC EQUATIONS GRAPH A CURVE GIVEN BY PARAMETRIC EQUATIONS ELIMINATE THE PARAMETER OF A PARAMETRIC EQUATION THE BRACHISTOCHRONE PROBLEM PARAMETRIC EQUATIONS AND PROJECTILE MOTION 6.4 Sequences, Series, and Summation Notation INFINITE SEQUENCES FACTORIALS ARITHMETIC SEQUENCES GEOMETRIC SEQUENCES PARTIAL SUMS AND SUMMATION NOTATION ARITHMETIC SERIES FINITE GEOMETRIC SERIES INFINITE GEOMETRIC SERIES 6.5 The Binomial Theorem BINOMIAL THEOREM ith TERM OF A BINOMIAL EXPANSION PASCAL’S TRIANGLE EXPLORING CONCEPTS WITH TECHNOLOGY: Using a Graphing Calculator to Find the nth Roots of z Chapter 6 Summary Chapter 6 True/False Exercises Chapter 6 Review Exercises Chapter 6 Test Cumulative Review Exercises ALGEBRA REVIEW APPENDIX A.1 Integer and Rational Number Exponents A.2 Polynomials A.3 Factoring SOLUTIONS TO THE TRY EXERCISES ANSWERS TO SELECTED EXERCISES INDEX