Annotated Instructor’s Edition PRECALCULUS PAUL SISSON THIRD EDITION © All Rights Reser ved.Cover Design: Patrick Thompson Cover Artwork: George Hart George Hart is a sculptor and applied mathematician who demonstrates how mathematics is cool and creative in ways you might not have expected. Whether he is slicing a bagel into two linked halves or leading hundreds of participants in an intricate geometric sculpture barn raising, he always finds original ways to share the beauty of mathematical thinking. Hart’s career includes eight years as a professor at Columbia University, fifteen years as a Research Professor at Stony Brook University, and five years cofounding the Museum of Mathematics in New York City. Now a full-time artist and consultant, he also makes videos that show the fun and creative sides of mathematics. See http://georgehart.com for examples of his work. Manager of Math Content Development: Chelsey Cooke Project Manager: Emily Christian A division of Quant Systems, Inc. 546 Long Point Road Mount Pleasant, SC 29464 Copyright © 2021 by Hawkes Learning / Quant Systems, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written consent of the publisher. Library of Congress Control Number 2020906521 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN: 978-1-64277-171-8 AIE ISBN: 978-1-64277-172-5 Editors: Danielle C. Bess, Daniel Breuer, S. Rebecca Johnson, Claudia Vance Assistant Editors: Sarah L. Allen, Allison Conger, Marvin Glover, Lisa Hinton Index: Barbara Miller Creative Services Manager: Trudy Gove Contributing Designers: Patrick Thompson, Joshua A. Walker Composition and Answer Key Assistance: Quant Systems India Pvt. Ltd. Courseware Developers: Vince Cellini, Adam Flaherty Technology Assistant: Kyle Gilstrap © All Rights Reser ved.Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Common Subsets of Real Numbers w The Real Number Line w Order on the Real Number Line w Set-Builder Notation and Interval Notation w Basic Set Operations and Venn Diagrams w Absolute Value and Distance on the Real Number Line w Components and Terminology of Algebraic Expressions w The Field Properties and Their Use in Algebra Natural Number and Integer Exponents w Properties of Exponents w Scientific Notation w Working with Geometric Formulas w Radical Notation w Simplifying and Combining Radical Expressions w Rational Number Exponents The Terminology of Polynomial Expressions w Basic Operations with Polynomials w Greatest Common Factor (GCF) w Factoring by Grouping w Factoring Special Binomials w Factoring Trinomials w Factoring Expressions Containing Noninteger Rational Exponents Simplifying Rational Expressions w Combining Rational Expressions w Simplifying Complex Rational Expressions The Imaginary Unit i and Its Properties w Basic Operations with Complex Numbers w Roots and Complex Numbers Equations and the Meaning of Solutions w Solving Linear Equations w Solving Linear Absolute Value Equations w Solving Linear Equations for One Variable w Applications of Linear Equations Solving Linear Inequalities w Solving Double Linear Inequalities w Solving Linear Absolute Value Inequalities w Applications of Linear Inequalities Solving Quadratic Equations by Factoring w Solving “Perfect Square” Quadratic Equations w Solving Quadratic Equations by Completing the Square w The Quadratic Formula w Applications of Quadratic Equations w Solving Quadratic-Like Equations w Solving General Polynomial Equations by Factoring w Solving Polynomial-Like Equations by Factoring Solving Rational Equations w Applications of Rational Equations w Solving Radical Equations w Solving Equations with Positive Rational Exponents w Solving Equations for One Variable Reser ved.iv Table of Contents 2 The Cartesian Coordinate System w Graphing Equations w The Distance and Midpoint Formulas The Standard Form of the Equation of a Circle w Graphing Circles Recognizing Linear Equations in Two Variables w Intercepts of the Coordinate Axes w Horizontal and Vertical Lines The Slope of a Line w The Slope-Intercept Form of the Equation of a Line w The Point-Slope Form of the Equation of a Line Slopes of Parallel Lines w Slopes of Perpendicular Lines Graphing Linear Inequalities w Graphing Linear Inequalities Joined by “And” or “Or” w Graphing Linear Absolute Value Inequalities 3 Relations, Domain, and Range w Functions and the Vertical Line Test w Function Notation and Function Evaluation w Implied Domain of a Function Linear Functions and Their Graphs w Linear Regression Quadratic Functions and Their Graphs w Quadratic Regression w Maximization/Minimization Problems Power Functions of the Form axn w Power Functions of the Form ax n- w Power Functions of the Form axn 1 w The Absolute Value Function w The Greatest Integer Function w Piecewise-Defined Functions Direct Variation w Inverse Variation w Joint Variation w Multivariable Functions Constructing Mathematical Models w Interpolation and Extrapolation © All Rights ved.Table of Contents v 4 Shifting Graphs Vertically and Horizontally w Reflecting Graphs w Stretching Graphs Vertically and Horizontally w Order of Transformations Symmetry of Functions and Equations w Intervals of Monotonicity w Local Extrema w Average Rate of Change Combining Functions Arithmetically w Composing Functions w Decomposing Functions w Recursive Graphics Inverses of Relations w Inverse Functions and the Horizontal Line Test w Finding the Inverse of a Function 5 Zeros of Polynomial Functions and Solutions of Polynomial Equations w Graphing Factored Polynomial Functions w Solving Polynomial Inequalities The Division Algorithm and the Remainder Theorem w Polynomial Long Division w Synthetic Division w Constructing Polynomials with Given Zeros The Rational Zero Theorem w Descartes’ Rule of Signs w Bounds of Real Zeros w The Intermediate Value Theorem The Fundamental Theorem of Algebra w Multiple Zeros and Their Geometric Meaning w Conjugate Pairs of Zeros w Summary of Polynomial Methods Characteristics of Rational Functions w Vertical Asymptotes w Horizontal and Oblique Asymptotes w Graphing Rational Functions w Solving Rational Inequalities © All vi Table of Contents 6 Characteristics of Exponential Functions w Graphing Exponential Functions w Solving Elementary Exponential Equations Models of Population Growth w Models of Radioactive Decay w Compound Interest and the Number e w Exponential Regression Characteristics of Logarithmic Functions w Graphing Logarithmic Functions w Evaluating Elementary Logarithmic Expressions w Solving Elementary Logarithmic Equations w Common and Natural Logarithms Properties of Logarithms w The Change of Base Formula w Applications of Logarithmic Functions w Logarithmic Regression Converting between Exponential and Logarithmic Forms w Applications of Exponential and Logarithmic Equations w Analysis of a Stock Market Investment 7 The Unit Circle and Angle Measure w Converting between Degrees and Radians w The Pythagorean Theorem and Commonly Encountered Angles w Arc Length and Angular Speed w Area of a Circular Sector The Trigonometric Functions w Evaluating Trigonometric Functions w Applications of Trigonometric Functions Extending the Domains of the Trigonometric Functions w Evaluating Trigonometric Functions Using Reference Angles w Relationships between Trigonometric Functions Graphing Sine and Cosine Functions w Periodicity and Symmetry w Amplitude, Frequency, and Phase Shifts w Simple Harmonic Motion w Damped Harmonic Motion Graphing Tangent and Cotangent Functions w Graphing Secant and Cosecant Functions The Inverse Trigonometric Functions w Evaluating Inverse Trigonometric Functions w Applications of Inverse Trigonometric Functions © All Rights ved.Table of Contents vii 8 Fundamental Trigonometric Identities w Simplifying Trigonometric Expressions w Verifying Trigonometric Identities w Trigonometric Substitutions Sum and Difference Identities w Using Sum and Difference Identities for Exact Evaluation w Using Sum and Difference Identities for Verification and Simplification Double-Angle Identities w Power-Reducing Identities w Half-Angle Identities w Product-to-Sum and Sum-to-Product Identities Solving Trigonometric Equations Using Algebraic Techniques w Solving Trigonometric Equations Using Inverse Trigonometric Functions 9 The Law of Sines w Applications of the Law of Sines w The Law of Sines and the Area of a Triangle The Law of Cosines w Applications of the Law of Cosines w Heron’s Formula for the Area of a Triangle The Polar Coordinate System w Converting between Polar and Cartesian Coordinates w The Form of Polar Equations w Graphing Polar Equations Applications of Parametric Equations w Graphing Parametric Equations by Eliminating the Parameter w Constructing Parametric Equations to Describe a Graph The Complex Plane w Complex Numbers in Trigonometric Form w Multiplying and Dividing Complex Numbers w Powers of Complex Numbers w Roots of Complex Numbers Vector Terminology w Basic Vector Operations w Component Form of a Vector w Applications of Vectors Properties of the Dot Product w Projections of Vectors w Applications of the Dot Product The Hyperbolic Functions w Hyperbolic Identities w The Inverse Hyperbolic Functions © All viii Table of Contents 10 Overview of Conic Sections w The Standard Form of the Equation of an Ellipse w Applications of Ellipses The Standard Form of the Equation of a Parabola (as a Conic Section) w Applications of Parabolas The Standard Form of the Equation of a Hyperbola w Applications of Hyperbolas Rotation Relations w Rotation Invariants The Focus and Directrix of a Conic Section w Graphing Polar Equations of Conic Sections 11 Systems of Linear Equations w Solving Systems of Linear Equations by Substitution w Solving Systems of Linear Equations by Elimination w Systems of Linear Equations in Three or More Variables w Applications of Systems of Linear Equations Systems of Linear Equations, Matrices, and Augmented Matrices w Gaussian Elimination and Row Echelon Form w Gauss-Jordan Elimination and Reduced Row Echelon Form Evaluating Determinants w Solving Systems of Linear Equations Using Cramer’s Rule Matrix Addition w Scalar Multiplication w Matrix Multiplication w Transition Matrices The Matrix Form of a System of Linear Equations w Finding the Inverse of a Matrix w Solving Systems of Linear Equations Using Matrix Inverses The Pattern of Partial Fraction Decompositions w Completing the Partial Fraction Decomposition Process Systems of Linear Inequalities w Planar Feasible Regions w Linear Programming in Two Variables Solving Systems of Nonlinear Equations by Graphing w Solving Systems of Nonlinear Equations Algebraically w Solving Systems of Nonlinear Inequalities © All Rights ReserTable of Contents ix 12 Recursively and Explicitly Defined Sequences w Summation Notation and Formulas w Partial Sums and Series w Fibonacci Sequences Arithmetic Sequences and Series w The Formula for the General Term of an Arithmetic Sequence w Evaluating Partial Sums of Arithmetic Sequences Geometric Sequences w The Formula for the General Term of a Geometric Sequence w Evaluating Partial Sums of Geometric Sequences w Evaluating Infinite Geometric Series w Zeno’s Paradoxes The Role of Induction w Proofs by Mathematical Induction The Multiplication Principle of Counting w Permutations w Combinations w The Binomial Theorem w The Multinomial Theorem The Language of Probability w Computing Probabilities Using Combinatorics w Unions, Intersections, and Independent Events 13 An Introduction to Limits, Continuity, and the Derivative 13 .1 Rates of Change and Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .971 The Difference Quotient w The Velocity Problem w The Tangent Problem 13 .2 Limits in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .984 Limits in Verbal, Numerical, and Visual Forms w Vertical Asymptotes and One-Sided Limits w Horizontal Asymptotes and Limits at Infinity 13 .3 The Mathematical Definition of Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1000 The Formal Definition of a Limit w Proving a Limit Exists w Proving a Limit Does Not Exist 13 .4 Determining Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1013 Limit Laws w Limit Determination Techniques 13 .5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1028 Continuity and Discontinuity at a Point w Continuous Functions w Properties of Continuity 13 .6 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1046 The Velocity and Tangent Problems Revisited w The Derivative as a Function Chapter 13 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1060 Chapter 13 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1061 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AK-1 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1 © All Rights ReserNext >