Thomas' Calculus Early Transcendentals PDF

Thomas’ Calculus Early Transcendentals Thirteenth Edition Based on the original work by George B. Thomas, Jr. Massachusetts Institute of Technology as revised by Maurice D. Weir Naval Postgraduate School Joel Hass University of California, Davis w...

Thomas’ Calculus Early Transcendentals Thirteenth Edition Based on the original work by George B. Thomas, Jr. Massachusetts Institute of Technology as revised by Maurice D. Weir Naval Postgraduate School Joel Hass University of California, Davis with the assistance of Christopher Heil Georgia Institute of Technology Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editor-in-Chief: Deirdre Lynch Senior Acquisitions Editor: William Hoffman Senior Content Editor: Rachel S. 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Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Pearson Education was aware of a trademark claim, the designa- tions have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data Weir, Maurice D. Thomas’ calculus : early transcendentals : based on the original work by George B. Thomas, Jr., Massachusetts Institute of Technology.—Thirteenth edition / as revised by Maurice D. Weir, Naval Postgraduate School, Joel Hass, University of California, Davis. pages cm ISBN 978-0-321-88407-7 (hardcover) I. Hass, Joel. II. Thomas, George B. (George Brinton), Jr., 1914–2006. Calculus. Based on (Work): III. Title. IV. Title: Calculus. QA303.2.W45 2014 515–dc23 2013023096 Copyright © 2014, 2010, 2008 Pearson Education, 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 permission of the publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-848-7047, or e-mail at http://www.pearsoned.com/legal/permissions.htm. 1 2 3 4 5 6 7 8 9 10—CRK—17 16 15 14 13 ISBN-10: 0-321-88407-8 ISBN-13: 978-0-321-88407-7 www.pearsonhighered.com  Contents Preface ix 1 Functions  1 1.1 Functions and Their Graphs 1 1.2 Combining Functions; Shifting and Scaling Graphs 14 1.3 Trigonometric Functions 21 1.4 Graphing with Software 29 1.5 Exponential Functions 36 1.6 Inverse Functions and Logarithms 41 Questions to Guide Your Review 54 Practice Exercises 54 Additional and Advanced Exercises 57 2 Limits and Continuity  59 2.1 Rates of Change and Tangents to Curves 59 2.2 Limit of a Function and Limit Laws 66 2.3 The Precise Definition of a Limit 77 2.4 One-Sided Limits 86 2.5 Continuity 93 2.6 Limits Involving Infinity; Asymptotes of Graphs 104 Questions to Guide Your Review 118 Practice Exercises 118 Additional and Advanced Exercises 120 3 Derivatives  123 3.1 Tangents and the Derivative at a Point 123 3.2 The Derivative as a Function 128 3.3 Differentiation Rules 136 3.4 The Derivative as a Rate of Change 146 3.5 Derivatives of Trigonometric Functions 156 3.6 The Chain Rule 163 3.7 Implicit Differentiation 171 3.8 Derivatives of Inverse Functions and Logarithms 177 3.9 Inverse Trigonometric Functions 187 3.10 Related Rates 193 3.11 Linearization and Differentials 202 Questions to Guide Your Review 214 Practice Exercises 215 Additional and Advanced Exercises 219 iii iv Contents 4 Applications of Derivatives  223 4.1 Extreme Values of Functions 223 4.2 The Mean Value Theorem 231 4.3 Monotonic Functions and the First Derivative Test 239 4.4 Concavity and Curve Sketching 244 4.5 Indeterminate Forms and L’Hôpital’s Rule 255 4.6 Applied Optimization 264 4.7 Newton’s Method 276 4.8 Antiderivatives 281 Questions to Guide Your Review 291 Practice Exercises 291 Additional and Advanced Exercises 295 5 Integrals  299 5.1 Area and Estimating with Finite Sums 299 5.2 Sigma Notation and Limits of Finite Sums 309 5.3 The Definite Integral 316 5.4 The Fundamental Theorem of Calculus 328 5.5 Indefinite Integrals and the Substitution Method 339 5.6 Definite Integral Substitutions and the Area Between Curves 347 Questions to Guide Your Review 357 Practice Exercises 357 Additional and Advanced Exercises 361 6 Applications of Definite Integrals  365 6.1 Volumes Using Cross-Sections 365 6.2 Volumes Using Cylindrical Shells 376 6.3 Arc Length 384 6.4 Areas of Surfaces of Revolution 390 6.5 Work and Fluid Forces 395 6.6 Moments and Centers of Mass 404 Questions to Guide Your Review 415 Practice Exercises 416 Additional and Advanced Exercises 417 7 Integrals and Transcendental Functions  420 7.1 The Logarithm Defined as an Integral 420 7.2 Exponential Change and Separable Differential Equations 430 7.3 Hyperbolic Functions 439 7.4 Relative Rates of Growth 448 Questions to Guide Your Review 453 Practice Exercises 453 Additional and Advanced Exercises 455  Contents v 8 Techniques of Integration  456 8.1 Using Basic Integration Formulas 456 8.2 Integration by Parts 461 8.3 Trigonometric Integrals 469 8.4 Trigonometric Substitutions 475 8.5 Integration of Rational Functions by Partial Fractions 480 8.6 Integral Tables and Computer Algebra Systems 489 8.7 Numerical Integration 494 8.8 Improper Integrals 504 8.9 Probability 515 Questions to Guide Your Review 528 Practice Exercises 529 Additional and Advanced Exercises 531 9 First-Order Differential Equations  536 9.1 Solutions, Slope Fields, and Euler’s Method 536 9.2 First-Order Linear Equations 544 9.3 Applications 550 9.4 Graphical Solutions of Autonomous Equations 556 9.5 Systems of Equations and Phase Planes 563 Questions to Guide Your Review 569 Practice Exercises 569 Additional and Advanced Exercises 570 10 Infinite Sequences and Series  572 10.1 Sequences 572 10.2 Infinite Series 584 10.3 The Integral Test 593 10.4 Comparison Tests 600 10.5 Absolute Convergence; The Ratio and Root Tests 604 10.6 Alternating Series and Conditional Convergence 610 10.7 Power Series 616 10.8 Taylor and Maclaurin Series 626 10.9 Convergence of Taylor Series 631 10.10 The Binomial Series and Applications of Taylor Series 638 Questions to Guide Your Review 647 Practice Exercises 648 Additional and Advanced Exercises 650 11 Parametric Equations and Polar Coordinates  653 11.1 Parametrizations of Plane Curves 653 11.2 Calculus with Parametric Curves 661 11.3 Polar Coordinates 671 vi Contents 11.4 Graphing Polar Coordinate Equations 675 11.5 Areas and Lengths in Polar Coordinates 679 11.6 Conic Sections 683 11.7 Conics in Polar Coordinates 692 Questions to Guide Your Review 699 Practice Exercises 699 Additional and Advanced Exercises 701 12 Vectors and the Geometry of Space  704 12.1 Three-Dimensional Coordinate Systems 704 12.2 Vectors 709 12.3 The Dot Product 718 12.4 The Cross Product 726 12.5 Lines and Planes in Space 732 12.6 Cylinders and Quadric Surfaces 740 Questions to Guide Your Review 745 Practice Exercises 746 Additional and Advanced Exercises 748 13 Vector-Valued Functions and Motion in Space  751 13.1 Curves in Space and Their Tangents 751 13.2 Integrals of Vector Functions; Projectile Motion 759 13.3 Arc Length in Space 768 13.4 Curvature and Normal Vectors of a Curve 772 13.5 Tangential and Normal Components of Acceleration 778 13.6 Velocity and Acceleration in Polar Coordinates 784 Questions to Guide Your Review 788 Practice Exercises 788 Additional and Advanced Exercises 790 14 Partial Derivatives  793 14.1 Functions of Several Variables 793 14.2 Limits and Continuity in Higher Dimensions 801 14.3 Partial Derivatives 810 14.4 The Chain Rule 821 14.5 Directional Derivatives and Gradient Vectors 830 14.6 Tangent Planes and Differentials 839 14.7 Extreme Values and Saddle Points 848 14.8 Lagrange Multipliers 857 14.9 Taylor’s Formula for Two Variables 866 14.10 Partial Derivatives with Constrained Variables 870 Questions to Guide Your Review 875 Practice Exercises 876 Additional and Advanced Exercises 879  Contents vii 15 Multiple Integrals  882 15.1 Double and Iterated Integrals over Rectangles 882 15.2 Double Integrals over General Regions 887 15.3 Area by Double Integration 896 15.4 Double Integrals in Polar Form 900 15.5 Triple Integrals in Rectangular Coordinates 906 15.6 Moments and Centers of Mass 915 15.7 Triple Integrals in Cylindrical and Spherical Coordinates 922 15.8 Substitutions in Multiple Integrals 934 Questions to Guide Your Review 944 Practice Exercises 944 Additional and Advanced Exercises 947 16 Integrals and Vector Fields  950 16.1 Line Integrals 950 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 957 16.3 Path Independence, Conservative Fields, and Potential Functions 969 16.4 Green’s Theorem in the Plane 980 16.5 Surfaces and Area 992 16.6 Surface Integrals 1003 16.7 Stokes’ Theorem 1014 16.8 The Divergence Theorem and a Unified Theory 1027 Questions to Guide Your Review 1039 Practice Exercises 1040 Additional and Advanced Exercises 1042 17 Second-Order Differential Equations  online 17.1 Second-Order Linear Equations 17.2 Nonhomogeneous Linear Equations 17.3 Applications 17.4 Euler Equations 17.5 Power Series Solutions Appendices  AP-1 A.1 Real Numbers and the Real Line AP-1 A.2 Mathematical Induction AP-6 A.3 Lines, Circles, and Parabolas AP-10 A.4 Proofs of Limit Theorems AP-19 A.5 Commonly Occurring Limits AP-22 A.6 Theory of the Real Numbers AP-23 A.7 Complex Numbers AP-26 A.8 The Distributive Law for Vector Cross Products AP-35 A.9 The Mixed Derivative Theorem and the Increment Theorem AP-36 Answers to Odd-Numbered Exercises  A-1 Index  I-1 Credits  C-1 A Brief Table of Integrals  T-1 This page intentionally left blank  Preface Thomas’ Calculus: Early Transcendentals, Thirteenth Edition, provides a modern intro- duction to calculus that focuses on conceptual understanding in developing the essential elements of a traditional course. This material supports a three-semester or four-quarter calculus sequence typically taken by students in mathematics, engineering, and the natural sciences. Precise explanations, thoughtfully chosen examples, superior figures, and time- tested exercise sets are the foundation of this text. We continue to improve this text in keeping with shifts in both the preparation and the ambitions of today’s students, and the applications of calculus to a changing world. Many of today’s students have been exposed to the terminology and computational methods of calculus in high school. Despite this familiarity, their acquired algebra and trigonometry skills sometimes limit their ability to master calculus at the college level. In this text, we seek to balance students’ prior experience in calculus with the algebraic skill development they may still need, without slowing their progress through calculus itself. We have taken care to provide enough review material (in the text and appendices), detailed solutions, and variety of examples and exercises, to support a complete understanding of calculus for students at varying levels. We present the material in a way to encourage stu- dent thinking, going beyond memorizing formulas and routine procedures, and we show students how to generalize key concepts once they are introduced. References are made throughout which tie a new concept to a related one that was studied earlier, or to a gen- eralization they will see later on. After studying calculus from Thomas, students will have developed problem solving and reasoning abilities that will serve them well in many im- portant aspects of their lives. Mastering this beautiful and creative subject, with its many practical applications across so many fields of endeavor, is its own reward. But the real gift of studying calculus is acquiring the ability to think logically and factually, and learning how to generalize conceptually. We intend this book to encourage and support those goals. New to this Edition In this new edition we further blend conceptual thinking with the overall logic and struc- ture of single and multivariable calculus. We continue to improve clarity and precision, taking into account helpful suggestions from readers and users of our previous texts. While keeping a careful eye on length, we have created additional examples throughout the text. Numerous new exercises have been added at all levels of difficulty, but the focus in this revision has been on the mid-level exercises. A number of figures have been reworked and new ones added to improve visualization. We have written a new section on probability, which provides an important application of integration to the life sciences. We have maintained the basic structure of the Table of Contents, and retained im- provements from the twelfth edition. In keeping with this process, we have added more improvements throughout, which we detail here: ix x Preface Functions In discussing the use of software for graphing purposes, we added a brief subsection on least squares curve fitting, which allows students to take advantage of this widely used and available application. Prerequisite material continues to be re- viewed in Appendices 1–3. Continuity We clarified the continuity definitions by confining the term “endpoints” to intervals instead of more general domains, and we moved the subsection on continuous extension of a function to the end of the continuity section. Derivatives We included a brief geometric insight justifying l’Hôpital’s Rule. We also enhanced and clarified the meaning of differentiability for functions of several vari- ables, and added a result on the Chain Rule for functions defined along a path. Integrals We wrote a new section reviewing basic integration formulas and the Sub- stitution Rule, using them in combination with algebraic and trigonometric identities, before presenting other techniques of integration. Probability We created a new section applying improper integrals to some commonly used probability distributions, including the exponential and normal distributions. Many examples and exercises apply to the life sciences. Series We now present the idea of absolute convergence before giving the Ratio and Root Tests, and then state these tests in their stronger form. Conditional convergence is introduced later on with the Alternating Series Test. Multivariable and Vector Calculus We give more geometric insight into the idea of multiple integrals, and we enhance the meaning of the Jacobian in using substitutions to evaluate them. The idea of surface integrals of vector fields now parallels the notion for line integrals of vector fields. We have improved our discussion of the divergence and curl of a vector field. Exercises and Examples Strong exercise sets are traditional with Thomas’ Calculus, and we continue to strengthen them with each new edition. Here, we have updated, changed, and added many new exercises and examples, with particular attention to in- cluding more applications to the life science areas and to contemporary problems. For instance, we updated an exercise on the growth of the U.S. GNP and added new exer- cises addressing drug concentrations and dosages, estimating the spill rate of a ruptured oil pipeline, and predicting rising costs for college tuition. Continuing Features RIGOR The level of rigor is consistent with that of earlier editions. We continue to distin- guish between formal and informal discussions and to point out their differences. We think starting with a more intuitive, less formal, approach helps students understand a new or dif- ficult concept so they can then appreciate its full mathematical precision and outcomes. We pay attention to defining ideas carefully and to proving theorems appropriate for calculus students, while mentioning deeper or subtler issues they would study in a more advanced course. Our organization and distinctions between informal and formal discussions give the instructor a degree of flexibility in the amount and depth of coverage of the various top- ics. For example, while we do not prove the Intermediate Value Theorem or the Extreme Value Theorem for continuous functions on a # x # b, we do state these theorems precisely, illustrate their meanings in numerous examples, and use them to prove other important re- sults. Furthermore, for those instructors who desire greater depth of coverage, in Appendix 6 we discuss the reliance of the validity of these theorems on the completeness of the real numbers.  Preface xi WRITING EXERCISES Writing exercises placed throughout the text ask students to ex- plore and explain a variety of calculus concepts and applications. In addition, the end of each chapter contains a list of questions for students to review and summarize what they have learned. Many of these exercises make good writing assignments. END-OF-CHAPTER REVIEWS AND PROJECTS In addition to problems appearing after each section, each chapter culminates with review questions, practice exercises covering the entire chapter, and a series of Additional and Advanced Exercises serving to include more challenging or synthesizing problems. Most chapters also include descriptions of several Technology Application Projects that can be worked by individual students or groups of students over a longer period of time. These projects require the use of a com- puter running Mathematica or Maple and additional material that is available over the Internet at www.pearsonhighered.com/thomas and in MyMathLab. WRITING AND APPLICATIONS As always, this text continues to be easy to read, conversa- tional, and mathematically rich. Each new topic is motivated by clear, easy-to-understand examples and is then reinforced by its application to real-world problems of immediate interest to students. A hallmark of this book has been the application of calculus to science and engineering. These applied problems have been updated, improved, and extended con- tinually over the last several editions. TECHNOLOGY In a course using the text, technology can be incorporated according to the taste of the instructor. Each section contains exercises requiring the use of technology; these are marked with a T if suitable for calculator or computer use, or they are labeled Computer Explorations if a computer algebra system (CAS, such as Maple or Math- ematica) is required. Additional Resources INSTRUCTOR’S SOLUTIONS MANUAL Single Variable Calculus (Chapters 1–11), ISBN 0-321-88408-6 | 978-0-321-88408-4 Multivariable Calculus (Chapters 10–16), ISBN 0-321-87901-5 | 978-0-321-87901-1 The Instructor’s Solutions Manual contains complete worked-out solutions to all of the exercises in Thomas’ Calculus: Early Transcendentals. STUDENT’S SOLUTIONS MANUAL Single Variable Calculus (Chapters 1–11), ISBN 0-321-88410-8 | 978-0-321-88410-7 Multivariable Calculus (Chapters 10–16), ISBN 0-321-87897-3 | 978-0-321-87897-7 The Student’s Solutions Manual is designed for the student and contains carefully worked-out solutions to all the odd-numbered exercises in Thomas’ Calculus: Early Transcendentals. JUST-IN-TIME ALGEBRA AND TRIGONOMETRY FOR EARLY TRANSCENDENTALS CALCULUS, Fourth Edition ISBN 0-321-67103-1 | 978-0-321-67103-5 Sharp algebra and trigonometry skills are critical to mastering calculus, and Just-in-Time Algebra and Trigonometry for Early Transcendentals Calculus by Guntram Mueller and Ronald I. Brent is designed to bolster these skills while students study calculus. As stu- dents make their way through calculus, this text is with them every step of the way, show- ing them the necessary algebra or trigonometry topics and pointing out potential problem spots. The easy-to-use table of contents has algebra and trigonometry topics arranged in the order in which students will need them as they study calculus. xii Preface Technology Resource Manuals Maple Manual by Marie Vanisko, Carroll College Mathematica Manual by Marie Vanisko, Carroll College TI-Graphing Calculator Manual by Elaine McDonald-Newman, Sonoma State University These manuals cover Maple 17, Mathematica 8, and the TI-83 Plus/TI-84 Plus and TI-89, respectively. Each manual provides detailed guidance for integrating a specific software package or graphing calculator throughout the course, including syntax and commands. These manuals are available to qualified instructors through the Thomas’ Calculus: Early Transcendentals Web site, www.pearsonhighered.com/thomas, and MyMathLab. WEB SITE www.pearsonhighered.com/thomas The Thomas’ Calculus: Early Transcendentals Web site contains the chapter on Second- Order Differential Equations, including odd-numbered answers, and provides the ex- panded historical biographies and essays referenced in the text. The Technology Resource Manuals and the Technology Application Projects, which can be used as projects by in- dividual students or groups of students, are also available. MyMathLab® Online Course (access code required) MyMathLab from Pearson is the world’s leading online resource in mathematics, integrat- ing interactive homework, assessment, and media in a flexible, easy-to-use format. MyMathLab delivers proven results in helping individual students succeed. MyMathLab has a consistently positive impact on the quality of learning in higher education math instruction. MyMathLab can be successfully implemented in any environment—lab-based, hybrid, fully online, traditional—and demonstrates the quan- tifiable difference that integrated usage makes in regard to student retention, subse- quent success, and overall achievement. MyMathLab’s comprehensive online gradebook automatically tracks your students’ re- sults on tests, quizzes, homework, and in the study plan. You can use the gradebook to quickly intervene if your students have trouble, or to provide positive feedback on a job well done. The data within MyMathLab are easily exported to a variety of spreadsheet programs, such as Microsoft Excel. You can determine which points of data you want to export, and then analyze the results to determine success. MyMathLab provides engaging experiences that personalize, stimulate, and measure learning for each student. “Getting Ready” chapter includes hundreds of exercises that address prerequisite skills in algebra and trigonometry. Each student can receive remediation for just those skills he or she needs help with. Exercises: The homework and practice exercises in MyMathLab are correlated to the exercises in the textbook, and they regenerate algorithmically to give students unlim- ited opportunity for practice and mastery. The software offers immediate, helpful feed- back when students enter incorrect answers. Multimedia Learning Aids: Exercises include guided solutions, sample problems, animations, Java™ applets, videos, and eText access for extra help at point-of-use. Expert Tutoring: Although many students describe the whole of MyMathLab as “like having your own personal tutor,” students using MyMathLab do have access to live tutoring from Pearson, from qualified math and statistics instructors.  Preface xiii And, MyMathLab comes from an experienced partner with educational expertise and an eye on the future. Knowing that you are using a Pearson product means knowing that you are using qual- ity content. It means that our eTexts are accurate and our assessment tools work. It also means we are committed to making MyMathLab as accessible as possible. Whether you are just getting started with MyMathLab, or have a question along the way, we’re here to help you learn about our technologies and how to incorporate them into your course. To learn more about how MyMathLab combines proven learning applications with power- ful assessment, visit www.mymathlab.com or contact your Pearson representative. Video Lectures with Optional Captioning The Video Lectures with Optional Captioning feature an engaging team of mathemat- ics instructors who present comprehensive coverage of topics in the text. The lecturers’ presentations include examples and exercises from the text and support an approach that emphasizes visualization and problem solving. Available only through MyMathLab and MathXL. MathXL® Online Course (access code required) MathXL® is the homework and assessment engine that runs MyMathLab. (MyMathLab is MathXL plus a learning management system.) With MathXL, instructors can: Create, edit, and assign online homework and tests using algorithmically generated ex- ercises correlated at the objective level to the textbook. Create and assign their own online exercises and import TestGen tests for added flexibility. Maintain records of all student work tracked in MathXL’s online gradebook. With MathXL, students can: Take chapter tests in MathXL and receive personalized study plans and/or personalized homework assignments based on their test results. Use the study plan and/or the homework to link directly to tutorial exercises for the objectives they need to study. Access supplemental animations and video clips directly from selected exercises. MathXL is available to qualified adopters. For more information, visit our website at www.mathxl.com, or contact your Pearson representative. TestGen® TestGen® (www.pearsoned.com/testgen) enables instructors to build, edit, print, and ad- minister tests using a computerized bank of questions developed to cover all the objec- tives of the text. TestGen is algorithmically based, allowing instructors to create multiple but equivalent versions of the same question or test with the click of a button. Instructors can also modify test bank questions or add new questions. The software and test bank are available for download from Pearson Education’s online catalog. PowerPoint® Lecture Slides These classroom presentation slides are geared specifically to the sequence and philosophy of the Thomas’ Calculus series. Key graphics from the book are included to help bring the concepts alive in the classroom.These files are available to qualified instructors through the Pearson Instructor Resource Center, www.pearsonhighered/irc, and MyMathLab. xiv Preface Acknowledgments We would like to express our thanks to the people who made many valuable contributions to this edition as it developed through its various stages: Accuracy Checkers Lisa Collette Patricia Nelson Tom Wegleitner Reviewers for Recent Editions Meighan Dillon, Southern Polytechnic State University Anne Dougherty, University of Colorado Said Fariabi, San Antonio College Klaus Fischer, George Mason University Tim Flood, Pittsburg State University Rick Ford, California State University—Chico Robert Gardner, East Tennessee State University Christopher Heil, Georgia Institute of Technology Joshua Brandon Holden, Rose-Hulman Institute of Technology Alexander Hulpke, Colorado State University Jacqueline Jensen, Sam Houston State University Jennifer M. Johnson, Princeton University Hideaki Kaneko, Old Dominion University Przemo Kranz, University of Mississippi Xin Li, University of Central Florida Maura Mast, University of Massachusetts—Boston Val Mohanakumar, Hillsborough Community College—Dale Mabry Campus Aaron Montgomery, Central Washington University Christopher M. Pavone, California State University at Chico Cynthia Piez, University of Idaho Brooke Quinlan, Hillsborough Community College—Dale Mabry Campus Rebecca A. Segal, Virginia Commonwealth University Andrew V. Sills, Georgia Southern University Alex Smith, University of Wisconsin—Eau Claire Mark A. Smith, Miami University Donald Solomon, University of Wisconsin—Milwaukee John Sullivan, Black Hawk College Maria Terrell, Cornell University Blake Thornton, Washington University in St. Louis David Walnut, George Mason University Adrian Wilson, University of Montevallo Bobby Winters, Pittsburg State University Dennis Wortman, University of Massachusetts—Boston  1 Functions OVERVIEW Functions are fundamental to the study of calculus. In this chapter we review what functions are and how they are pictured as graphs, how they are combined and trans- formed, and ways they can be classified. We review the trigonometric functions, and we discuss misrepresentations that can occur when using calculators and computers to obtain a function’s graph. We also discuss inverse, exponential, and logarithmic functions. The real number system, Cartesian coordinates, straight lines, circles, parabolas, and ellipses are reviewed in the Appendices. 1.1 Functions and Their Graphs Functions are a tool for describing the real world in mathematical terms. A function can be represented by an equation, a graph, a numerical table, or a verbal description; we will use all four representations throughout this book. This section reviews these function ideas. Functions; Domain and Range The temperature at which water boils depends on the elevation above sea level (the boiling point drops as you ascend). The interest paid on a cash investment depends on the length of time the investment is held. The area of a circle depends on the radius of the circle. The dis- tance an object travels at constant speed along a straight-line path depends on the elapsed time. In each case, the value of one variable quantity, say y, depends on the value of another variable quantity, which we might call x. We say that “y is a function of x” and write this symbolically as y = ƒ(x) (“y equals ƒ of x”). In this notation, the symbol ƒ represents the function, the letter x is the independent variable representing the input value of ƒ, and y is the dependent variable or output value of ƒ at x. DEFINITION A function ƒ from a set D to a set Y is a rule that assigns a unique (single) element ƒ(x) ∊Y to each element x∊D. The set D of all possible input values is called the domain of the function. The set of all output values of ƒ(x) as x varies throughout D is called the range of the function. The range may not include every element in the set Y. The domain and range of a function can be any sets of objects, but often in calculus they are sets of real numbers interpreted as points of a coordinate line. (In Chapters 13–16, we will encounter functions for which the elements of the sets are points in the coordinate plane or in space.) 1 2 Chapter 1: Functions Often a function is given by a formula that describes how to calculate the output value from the input variable. For instance, the equation A = pr 2 is a rule that calculates the area A of a circle from its radius r (so r, interpreted as a length, can only be positive in this formula). When we define a function y = ƒ(x) with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of real x-values for which the formula gives real y-values, which is called the natural domain. If we want to restrict the domain in some way, we must say so. The domain of y = x2 is the entire set of real numbers. To restrict the domain of the function to, say, positive values of x, we would write “y = x2, x 7 0.” Changing the domain to which we apply a formula usually changes the range as well. The range of y = x2 is [0, q). The range of y = x2, x Ú 2, is the set of all numbers obtained by squaring numbers greater than or equal to 2. In set notation (see Appendix 1), the range is 5x2  x Ú 26 or 5y  y Ú 46 or 3 4, q). x f f (x) Input Output (domain) (range) When the range of a function is a set of real numbers, the function is said to be real- FIGURE 1.1 A diagram showing a valued. The domains and ranges of most real-valued functions of a real variable we con- function as a kind of machine. sider are intervals or combinations of intervals. The intervals may be open, closed, or half open, and may be finite or infinite. Sometimes the range of a function is not easy to find. A function ƒ is like a machine that produces an output value ƒ(x) in its range whenever we feed it an input value x from its domain (Figure 1.1). The function keys on a calculator give an example of a function as a machine. For instance, the 2x key on a calculator gives an output x value (the square root) whenever you enter a nonnegative number x and press the 2x key. a f (a) f(x) A function can also be pictured as an arrow diagram (Figure 1.2). Each arrow associates an element of the domain D with a unique or single element in the set Y. In Figure 1.2, the D = domain set Y = set containing arrows indicate that ƒ(a) is associated with a, ƒ(x) is associated with x, and so on. Notice that the range a function can have the same value at two different input elements in the domain (as occurs FIGURE 1.2 A function from a set D with ƒ(a) in Figure 1.2), but each input element x is assigned a single output value ƒ(x). to a set Y assigns a unique element of Y to each element in D. EXAMPLE 1 Let’s verify the natural domains and associated ranges of some simple functions. The domains in each case are the values of x for which the formula makes sense. Function Domain (x) Range ( y) y = x2 (- q, q) 3 0, q) y = 1>x (- q, 0) ∪ (0, q) (- q, 0) ∪ (0, q) y = 2x 3 0, q) 3 0, q) y = 24 - x (- q, 44 3 0, q) y = 21 - x2 3 -1, 14 3 0, 14 Solution The formula y = x2 gives a real y-value for any real number x, so the domain is (- q, q). The range of y = x2 is 3 0, q) because the square of any real number is non- negative and every nonnegative number y is the square of its own square root, y = 1 2y 2 2 for y Ú 0. The formula y = 1>x gives a real y-value for every x except x = 0. For consistency in the rules of arithmetic, we cannot divide any number by zero. The range of y = 1>x, the set of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since y = 1>(1>y). That is, for y ≠ 0 the number x = 1>y is the input assigned to the output value y. The formula y = 2x gives a real y-value only if x Ú 0. The range of y = 2x is 3 0, q) because every nonnegative number is some number’s square root (namely, it is the square root of its own square). In y = 24 - x, the quantity 4 - x cannot be negative. That is, 4 - x Ú 0, or x … 4. The formula gives real y-values for all x … 4. The range of 24 - x is 3 0, q), the set of all nonnegative numbers.  1.1 Functions and Their Graphs 3 The formula y = 21 - x2 gives a real y-value for every x in the closed interval from -1 to 1. Outside this domain, 1 - x2 is negative and its square root is not a real number. The values of 1 - x2 vary from 0 to 1 on the given domain, and the square roots of these values do the same. The range of 21 - x2 is 3 0, 14. Graphs of Functions If ƒ is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for ƒ. In set notation, the graph is 5(x, ƒ(x))  x∊D6. The graph of the function ƒ(x) = x + 2 is the set of points with coordinates (x, y) for which y = x + 2. Its graph is the straight line sketched in Figure 1.3. The graph of a function ƒ is a useful picture of its behavior. If (x, y) is a point on the graph, then y = ƒ(x) is the height of the graph above (or below) the point x. The height may be positive or negative, depending on the sign of ƒ(x) (Figure 1.4). y y f (1) f (2) x y=x+2 x 0 1 2 2 f(x) (x, y) x y = x2 −2 0 x -2 4 FIGURE 1.3 The graph of ƒ(x) = x + 2 FIGURE 1.4 If (x, y) lies on the graph of -1 1 is the set of points (x, y) for which y has the ƒ, then the value y = ƒ(x) is the height of 0 0 value x + 2. the graph above the point x (or below x if 1 1 ƒ(x) is negative). 3 9 2 4 EXAMPLE 2 Graph the function y = x2 over the interval 3 -2, 24. 2 4 Solution Make a table of xy-pairs that satisfy the equation y = x2. Plot the points (x, y) whose coordinates appear in the table, and draw a smooth curve (labeled with its equation) through the plotted points (see Figure 1.5). y How do we know that the graph of y = x2 doesn’t look like one of these curves? (−2, 4) (2, 4) 4 y y y = x2 3 3 9 2 a2 , 4b (−1, 1) (1, 1) y = x 2? y = x 2? 1 x −2 −1 0 1 2 FIGURE 1.5 Graph of the function x x in Example 2. 4 Chapter 1: Functions To find out, we could plot more points. But how would we then connect them? The basic question still remains: How do we know for sure what the graph looks like between the points we plot? Calculus answers this question, as we will see in Chapter 4. Meanwhile, we will have to settle for plotting points and connecting them as best we can. Representing a Function Numerically We have seen how a function may be represented algebraically by a formula (the area function) and visually by a graph (Example 2). Another way to represent a function is numerically, through a table of values. Numerical representations are often used by engi- neers and experimental scientists. From an appropriate table of values, a graph of the func- tion can be obtained using the method illustrated in Example 2, possibly with the aid of a computer. The graph consisting of only the points in the table is called a scatterplot. EXAMPLE 3 Musical notes are pressure waves in the air. The data associated with Figure 1.6 give recorded pressure displacement versus time in seconds of a musical note produced by a tuning fork. The table provides a representation of the pressure function over time. If we first make a scatterplot and then connect approximately the data points (t, p) from the table, we obtain the graph shown in the figure. p (pressure) Time Pressure Time Pressure 1.0 Data 0.00091 -0.080 0.00362 0.217 0.8 0.6 0.00108 0.200 0.00379 0.480 0.4 0.00125 0.480 0.00398 0.681 0.2 t (sec) 0.00144 0.693 0.00416 0.810 0.001 0.002 0.003 0.004 0.005 0.006 −0.2 0.00162 0.816 0.00435 0.827 −0.4 −0.6 0.00180 0.844 0.00453 0.749 0.00198 0.771 0.00471 0.581 FIGURE 1.6 A smooth curve through the plotted points 0.00216 0.603 0.00489 0.346 gives a graph of the pressure function represented by the 0.00234 0.368 0.00507 0.077 accompanying tabled data (Example 3). 0.00253 0.099 0.00525 -0.164 0.00271 -0.141 0.00543 -0.320 0.00289 -0.309 0.00562 -0.354 0.00307 -0.348 0.00579 -0.248 0.00325 -0.248 0.00598 -0.035 0.00344 -0.041 The Vertical Line Test for a Function Not every curve in the coordinate plane can be the graph of a function. A function ƒ can have only one value ƒ(x) for each x in its domain, so no vertical line can intersect the graph of a function more than once. If a is in the domain of the function ƒ, then the vertical line x = a will intersect the graph of ƒ at the single point (a, ƒ(a)). A circle cannot be the graph of a function, since some vertical lines intersect the circle twice. The circle graphed in Figure 1.7a, however, does contain the graphs of functions of x, such as the upper semicircle defined by the function ƒ(x) = 21 - x2 and the lower semicircle defined by the function g(x) = - 21 - x2 (Figures 1.7b and 1.7c).  1.1 Functions and Their Graphs 5 y y y −1 1 x x x −1 0 1 −1 0 1 0 (a) x 2 + y 2 = 1 (b) y = "1 − x 2 (c) y = −"1 − x 2 FIGURE 1.7 (a) The circle is not the graph of a function; it fails the vertical line test. (b) The upper semicircle is the graph of a function ƒ(x) = 21 - x2. (c) The lower semicircle is the graph of a function g(x) = - 21 - x2. y y = 0x0 y = −x 3 y=x Piecewise-Defined Functions 2 1 Sometimes a function is described in pieces by using different formulas on different parts of its domain. One example is the absolute value function x −3 −2 −1 0 1 2 3 0x0 = e x, x Ú 0 First formula FIGURE 1.8 The absolute value function has domain (- q, q) and -x, x 6 0, Second formula range 30, q). whose graph is given in Figure 1.8. The right-hand side of the equation means that the function equals x if x Ú 0, and equals -x if x 6 0. Piecewise-defined functions often y arise when real-world data are modeled. Here are some other examples. y = −x y = f (x) 2 1 y=1 EXAMPLE 4 The function y = x2 x −2 −1 0 1 2 -x, x 6 0 First formula ƒ(x) = c x2, 0 … x … 1 Second formula FIGURE 1.9 To graph the 1, x 7 1 Third formula function y = ƒ(x) shown here, we apply different formulas to is defined on the entire real line but has values given by different formulas, depending on different parts of its domain the position of x. The values of ƒ are given by y = -x when x 6 0, y = x2 when (Example 4). 0 … x … 1, and y = 1 when x 7 1. The function, however, is just one function whose domain is the entire set of real numbers (Figure 1.9). y y=x 3 EXAMPLE 5 The function whose value at any number x is the greatest integer less than or equal to x is called the greatest integer function or the integer floor function. It is denoted : x ;. Figure 1.10 shows the graph. Observe that 2 y = :x; 1 −2 −1 x : 2.4 ; = 2, : 1.9 ; = 1, : 0 ; = 0, : -1.2 ; = -2, : 2 ; = 2, : 0.2 ; = 0, : -0.3 ; = -1, : -2 ; = -2. 1 2 3 −2 FIGURE 1.10 The graph of the EXAMPLE 6 The function whose value at any number x is the smallest integer greatest integer function y = : x ; greater than or equal to x is called the least integer function or the integer ceiling func- lies on or below the line y = x, so tion. It is denoted < x =. Figure 1.11 shows the graph. For positive values of x, this function it provides an integer floor for x might represent, for example, the cost of parking x hours in a parking lot that charges $1 (Example 5). for each hour or part of an hour. 6 Chapter 1: Functions y Increasing and Decreasing Functions y=x If the graph of a function climbs or rises as you move from left to right, we say that the 3 function is increasing. If the graph descends or falls as you move from left to right, the 2 function is decreasing. y = x, then sometimes it is said that y is inversely proportional to x (because 1>x is the multiplicative inverse of x). Power Functions A function ƒ(x) = xa, where a is a constant, is called a power function. There are several important cases to consider. 8 Chapter 1: Functions (a) a = n, a positive integer. The graphs of ƒ(x) = xn, for n = 1, 2, 3, 4, 5, are displayed in Figure 1.15. These func- tions are defined for all real values of x. Notice that as the power n gets larger, the curves tend to flatten toward the x-axis on the interval (-1, 1), and to rise more steeply for 0 x 0 7 1. Each curve passes through the point (1, 1) and through the origin. The graphs of functions with even powers are symmetric about the y-axis; those with odd powers are symmetric about the origin. The even-powered functions are decreasing on the interval (- q, 04 and increasing on 3 0, q); the odd-powered functions are increasing over the entire real line (- q, q). y y=x y y y y y = x5 y = x2 y = x3 y = x4 1 1 1 1 1 x x x x x −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 −1 −1 −1 −1 FIGURE 1.15 Graphs of ƒ(x) = xn, n = 1, 2, 3, 4, 5, defined for - q 6 x 6 q. (b) a = -1 or a = -2. The graphs of the functions ƒ(x) = x-1 = 1>x and g(x) = x-2 = 1>x2 are shown in Figure 1.16. Both functions are defined for all x ≠ 0 (you can never divide by zero). The graph of y = 1>x is the hyperbola xy = 1, which approaches the coordinate axes far from the origin. The graph of y = 1>x2 also approaches the coordinate axes. The graph of the function ƒ is symmetric about the origin; ƒ is decreasing on the intervals (- q, 0) and (0, q). The graph of the function g is symmetric about the y-axis; g is increasing on (- q, 0) and decreasing on (0, q). y y y = 1x y = 12 x 1 x 0 1 1 Domain: x ≠ 0 x Range: y ≠ 0 0 1 Domain: x ≠ 0 Range: y > 0 (a) (b) FIGURE 1.16 Graphs of the power functions ƒ(x) = xa for part (a) a = - 1 and for part (b) a = -2. 1 1 3 2 (c) a = , , , and. 2 3 2 3 3 The functions ƒ(x) = x1>2 = 2x and g(x) = x1>3 = 2 x are the square root and cube root functions, respectively. The domain of the square root function is 3 0, q), but the cube root function is defined for all real x. Their graphs are displayed in Figure 1.17, along with the graphs of y = x3>2 and y = x2>3. (Recall that x3>2 = (x1>2)3 and x2>3 = (x1>3)2.) Polynomials A function p is a polynomial if p(x) = an xn + an - 1xn - 1 + g + a1 x + a0 where n is a nonnegative integer and the numbers a0, a1, a2, c, an are real constants (called the coefficients of the polynomial). All polynomials have domain (- q, q). If the  1.1 Functions and Their Graphs 9 y y y y y = !x y=x 32 y = !x 3 y = x 23 1 1 1 1 x x x x 0 1 0 1 0 1 0 1 Domain: 0 ≤ x < ∞ Domain: −∞ < x < ∞ Domain: 0 ≤ x < ∞ Domain: −∞ < x < ∞ Range: 0 ≤ y < ∞ Range: −∞ < y < ∞ Range: 0 ≤ y < ∞ Range: 0 ≤ y < ∞ 1 1 3 2 FIGURE 1.17 Graphs of the power functions ƒ(x) = xa for a = , , , and. 2 3 2 3 leading coefficient an ≠ 0 and n 7 0, then n is called the degree of the polynomial. Lin- ear functions with m ≠ 0 are polynomials of degree 1. Polynomials of degree 2, usually written as p(x) = ax2 + bx + c, are called quadratic functions. Likewise, cubic functions are polynomials p(x) = ax3 + bx2 + cx + d of degree 3. Figure 1.18 shows the graphs of three polynomials. Techniques to graph polynomials are studied in Chapter 4. 3 2 y = x − x − 2x + 1 3 2 3 y 4 y y y = (x − 2)4(x + 1)3(x − 1) y= 8x 4 − 14x 3 − 9x 2 + 11x − 1 16 2 2 x −1 1 2 x −2 −4 −2 0 2 4 −4 x −1 0 1 2 −6 −2 −8 −10 −16 −4 −12 (a) (b) (c) FIGURE 1.18 Graphs of three polynomial functions. Rational Functions A rational function is a quotient or ratio ƒ(x) = p(x)>q(x), where p and q are polynomials. The domain of a rational function is the set of all real x for which q(x) ≠ 0. The graphs of several rational functions are shown in Figure 1.19. y y 8 y = 11x3 + 2 y = 5x +2 8x − 3 y 2 4 6 2x − 1 3x + 2 2 4 y = 2x − 3 2 2 7x + 4 1 Line y = 5 3 2 x x x −4 −2 2 4 −5 0 5 10 −4 −2 0 2 4 6 −1 −2 −2 −4 −2 NOT TO SCALE −4 −6 −8 (a) (b) (c) FIGURE 1.19 Graphs of three rational functions. The straight red lines approached by the graphs are called asymptotes and are not part of the graphs. We discuss asymptotes in Section 2.6. 10 Chapter 1: Functions Algebraic Functions Any function constructed from polynomials using algebraic oper- ations (addition, subtraction, multiplication, division, and taking roots) lies within the class of algebraic functions. All rational functions are algebraic, but also included are more complicated functions (such as those satisfying an equation like y3 - 9xy + x3 = 0, studied in Section 3.7). Figure 1.20 displays the graphs of three algebraic functions. y y = x 13(x − 4) y y = x(1 − x)25 4 y = 3 (x 2 − 1) 23 4 3 y 2 1 1 x x x −1 4 −1 0 1 0 5 1 −1 7 −2 −1 −3 (a) (b) (c) FIGURE 1.20 Graphs of three algebraic functions. Trigonometric Functions The six basic trigonometric functions are reviewed in Section 1.3. The graphs of the sine and cosine functions are shown in Figure 1.21. y y 1 1 3p 5p 3p − p2 2 2 x x −p 0 p 2p 0 p −1 −1 2 (a) f (x) = sin x (b) f (x) = cos x FIGURE 1.21 Graphs of the sine and cosine functions. Exponential Functions Functions of the form ƒ(x) = ax, where the base a 7 0 is a positive constant and a ≠ 1, are called exponential functions. All exponential functions have domain (- q, q) and range (0, q), so an exponential function never assumes the value 0. We discuss exponential functions in Section 1.5. The graphs of some exponential functions are shown in Figure 1.22. y y y = 10 x y = 10 –x 12 12 10 10 8 8 6 y= 3 –x 6 y = 3x 4 4 2 2 y = 2x y = 2 –x x x −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 (a) (b) FIGURE 1.22 Graphs of exponential functions.  1.1 Functions and Their Graphs 11 Logarithmic Functions These are the functions ƒ(x) = loga x, where the base a ≠ 1 is a positive constant. They are the inverse functions of the exponential functions, and we discuss these functions in Section 1.6. Figure 1.23 shows the graphs of four loga- rithmic functions with various bases. In each case the domain is (0, q) and the range is (- q, q). y y y = log 2 x y = log 3 x 1 x 0 1 y = log5 x 1 −1 y = log10 x x −1 0 1 FIGURE 1.23 Graphs of four logarithmic FIGURE 1.24 Graph of a catenary or functions. hanging cable. (The Latin word catena means “chain.”) Transcendental Functions These are functions that are not algebraic. They include the trigonometric, inverse trigonometric, exponential, and logarithmic functions, and many other functions as well. A particular example of a transcendental function is a catenary. Its graph has the shape of a cable, like a telephone line or electric cable, strung from one support to another and hanging freely under its own weight (Figure 1.24). The function defining the graph is discussed in Section 7.3. Exercises 1.1 Functions 8. a. y b. y In Exercises 1–6, find the domain and range of each function. 1. ƒ(x) = 1 + x2 2. ƒ(x) = 1 - 2x 3. F(x) = 25x + 10 4. g(x) = 2x2 - 3x 4 2 5. ƒ(t) = 6. G(t) = 3 - t t 2 - 16 In Exercises 7 and 8, which of the graphs are graphs of functions of x, x x and which are not? Give reasons for your answers. 0 0 7. a. y b. y Finding Formulas for Functions 9. Express the area and perimeter of an equilateral triangle as a function of the triangle’s side length x. 10. Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length. x x 11. Express the edge length of a cube as a function of the cube’s 0 0 diagonal length d. Then express the surface area and volume of the cube as a function of the diagonal length. 12 Chapter 1: Functions 12. A point P in the first quadrant lies on the graph of the function 31. a. y b. y ƒ(x) = 2x. Express the coordinates of P as functions of the slope of the line joining P to the origin. (−1, 1) (1, 1) 2 1 13. Consider the point (x, y) lying on the graph of the line x 2x + 4y = 5. Let L be the distance from the point (x, y) to the 3 1 x origin (0, 0). Write L as a function of x. (−2, −1) (1, −1) (3, −1) 14. Consider the point (x, y) lying on the graph of y = 2x - 3. Let L be the distance between the points (x, y) and (4, 0). Write L as a 32. a. y b. y function of y. (T, 1) 1 Functions and Graphs A Find the natural domain and graph the functions in Exercises 15–20. t 15. ƒ(x) = 5 - 2x 16. ƒ(x) = 1 - 2x - x2 0 T T 3T 2T 17. g(x) = 2 0 x 0 2 2 18. g(x) = 2- x x −A 19. F(t) = t> 0 t 0 20. G(t) = 1> 0 t 0 0 T T 2 x + 3 21. Find the domain of y =. 4 - 2x2 - 9 The Greatest and Least Integer Functions x2 33. For what values of x is 22. Find the range of y = 2 + 2 x + 4. a. : x ; = 0? b. < x = = 0? 23. Graph the following equations and explain why they are not 34. What real numbers x satisfy the equation : x ; = < x = ? graphs of functions of x. 35. Does < - x = = - : x ; for all real x? Give reasons for your answer. a. 0 y 0 = x b. y2 = x2 36. Graph the function 24. Graph the following equations and explain why they are not : x ;, x Ú 0 ƒ(x) = e x, x, 0 … x 41. y = 2 0 x 0 42. y = 2- x 43. y = x3 >8 44. y = - 4 2x Find a formula for each function graphed in Exercises 29–32. 45. y = - x3>2 46. y = (- x)2>3 29. a. y b. y Even and Odd Functions (1, 1) In Exercises 47–58, say whether the function is even, odd, or neither. 1 2 Give reasons for your answer. 47. ƒ(x) = 3 48. ƒ(x) = x-5 x t 0 2 0 1 2 3 4 49. ƒ(x) = x2 + 1 50. ƒ(x) = x2 + x 51. g(x) = x + x 3 52. g(x) = x4 + 3x2 - 1 30. a. y b. y 1 x 53. g(x) = 54. g(x) = 2 3 x2 - 1 x2 - 1 (2, 1) 2 1 55. h(t) = 56. h(t) =  t 3  2 5 x t - 1 1 x 57. h(t) = 2t + 1 58. h(t) = 2  t  + 1 −1 1 2 −1 (2, −1) Theory and Examples −2 59. The variable s is proportional to t, and s = 25 when t = 75. −3 Determine t when s = 60.  1.1 Functions and Their Graphs 13 60. Kinetic energy The kinetic energy K of a mass is proportional 66. a. y = 5x b. y = 5x c. y = x5 to the square of its velocity y. If K = 12,960 joules when y = 18 m>sec, what is K when y = 10 m>sec? y 61. The variables r and s are inversely proportional, and r = 6 when g s = 4. Determine s when r = 10. 62. Boyle’s Law Boyle’s Law says that the volume V of a gas at constant temperature increases whenever the pressure P decreases, so that V and P are inversely proportional. If P = 14.7 lb>in2 h when V = 1000 in3, then what is V when P = 23.4 lb>in2? x 0 63. A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 14 in. by 22 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a func- f tion of x. 22 T 67. a. Graph the functions ƒ(x) = x>2 and g(x) = 1 + (4>x) to- x x gether to identify the values of x for which x x x 4 14