Goldstein calculus and its applications 13th ed 2014 text

Thirteenth Edition CALCULUS & ITS APPLICATIONS Larry J Goldstein Goldstein Educational Technologies David C Lay University of Maryland David I Schneider University of Maryland Nakhl´e H Asmar University of Missouri Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montr´ eal Toronto Delhi Mexico City S˜ ao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editorial Director: Christine Hoag Editor in Chief: Deirdre Lynch Executive Editor: Jennifer Crum Editorial Assistant: Joanne Wendelken Executive Marketing Manager: Jeff Weidenaar Marketing Assistant: Caitlin Crain Executive Content Editor: Christine O’Brien Senior Managing Editor: Karen Wernholm Senior Production Supervisor: Ron Hampton Associate Design Director, Andrea Nix Interior Design: Cenveo Publisher Services Art Director/Cover Designer: Beth Paquin Composition and Project Management: Aptara, Inc Senior Technical Art Specialist: Joe Vetere Procurement Manager: Evelyn M Beaton Procurement Specialist: Debbie Rossi Media Producer: Jean Choe Software Development: Eileen Moore, MathXL; Marty Wright, TestGen Cover Image: Origami designed and folded by Sipho Mabona 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 was aware of a trademark claim, the designations have been printed in initial caps or all caps Library of Congress Cataloging-in-Publication Data Calculus and its applications.—13th ed / Larry J Goldstein [et al.] p cm Includes bibliographical references and index ISBN 0-321-84890-X Calculus—Textbooks I Goldstein, Larry Joel II Title: Calculus and its applications QA303.2.G66 2014 515—dc23 2012021184 Copyright c 2014, 2010, 2007 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, Boston, MA 02116 7—CRK—16 15 14 13 12 dumperina ISBN-13: 978-0-321-84890-1 ISBN-10: 0-321-84890-X Contents Preface vii Prerequisite Skills Diagnostic Test Introduction xv Functions 0.1 0.2 0.3 0.4 0.5 0.6 Functions and Their Graphs Some Important Functions 13 The Algebra of Functions 21 Zeros of Functions—The Quadratic Formula and Factoring 26 Exponents and Power Functions 33 Functions and Graphs in Applications 40 Chapter Summary and Chapter Review Exercises The Derivative 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 56 The Slope of a Straight Line 57 The Slope of a Curve at a Point 66 The Derivative and Limits 73 Limits and the Derivative 82 Differentiability and Continuity 92 Some Rules for Differentiation 98 More about Derivatives 104 The Derivative as a Rate of Change 112 Chapter Summary and Chapter Review Exercises 50 123 Applications of the Derivative 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Describing Graphs of Functions 131 131 The First- and Second-Derivative Rules 141 The First- and Second-Derivative Tests and Curve Sketching 149 Curve Sketching (Conclusion) 159 Optimization Problems 164 Further Optimization Problems 172 Applications of Derivatives to Business and Economics Chapter Summary and Chapter Review Exercises 180 189 iii iv CONTENTS Techniques of Differentiation 3.1 3.2 3.3 The Product and Quotient Rules 197 197 The Chain Rule and the General Power Rule Implicit Differentiation and Related Rates 206 212 Chapter Summary and Chapter Review Exercises The Exponential and Natural Logarithm Functions 4.1 4.2 4.3 4.4 4.5 4.6 Exponential Functions The Exponential Function ex 230 Differentiation of Exponential Functions The Natural Logarithm Function 235 240 The Derivative of ln x 244 Properties of the Natural Logarithm Function 247 256 Exponential Growth and Decay 257 Compound Interest 265 Applications of the Natural Logarithm Function to Economics 271 Further Exponential Models 278 Chapter Summary and Chapter Review Exercises 287 The Definite Integral 6.1 6.2 6.3 6.4 6.5 291 Antidifferentiation 292 The Definite Integral and Net Change of a Function The Definite Integral and Area under a Graph Areas in the xy-Plane 300 308 318 Applications of the Definite Integral 331 Chapter Summary and Chapter Review Exercises 251 Applications of the Exponential and Natural Logarithm Functions 5.1 5.2 5.3 5.4 339 Functions of Several Variables 7.1 7.2 7.3 7.4 7.5 7.6 226 226 Chapter Summary and Chapter Review Exercises 221 Examples of Functions of Several Variables 347 347 Partial Derivatives 353 Maxima and Minima of Functions of Several Variables Lagrange Multipliers and Constrained Optimization The Method of Least Squares Double Integrals 361 368 376 382 Chapter Summary and Chapter Review Exercises 386 CONTENTS The Trigonometric Functions 8.1 8.2 8.3 8.4 392 Radian Measure of Angles 392 The Sine and the Cosine 395 Differentiation and Integration of sin t and cos t 401 The Tangent and Other Trigonometric Functions 409 Chapter Summary and Chapter Review Exercises Integration by Substitution Integration by Parts 418 419 425 Evaluation of Definite Integrals 429 Approximation of Definite Integrals Some Applications of the Integral Improper Integrals 432 442 446 Chapter Summary and Chapter Review Exercises 10 458 Solutions of Differential Equations Separation of Variables 458 465 First-Order Linear Differential Equations 473 Applications of First-Order Linear Differential Equations Graphing Solutions of Differential Equations Applications of Differential Equations 477 484 492 Numerical Solution of Differential Equations 501 Chapter Summary and Chapter Review Exercises 506 Taylor Polynomials and Infinite Series 11.1 11.2 11.3 11.4 11.5 Taylor Polynomials 513 513 The Newton–Raphson Algorithm 520 Infinite Series 527 Series with Positive Terms 534 Taylor Series 540 Chapter Summary and Chapter Review Exercises 12 452 Differential Equations 10.1 10.2 10.3 10.4 10.5 10.6 10.7 11 413 Techniques of Integration 9.1 9.2 9.3 9.4 9.5 9.6 546 Probability and Calculus 12.1 12.2 12.3 12.4 12.5 v Discrete Random Variables 552 552 Continuous Random Variables 558 Expected Value and Variance 566 Exponential and Normal Random Variables Poisson and Geometric Random Variables 571 579 Chapter Summary and Chapter Review Exercises 586 vi CONTENTS Appendix Areas under the Standard Normal Curve Learning Objectives A-2 Sources S-1 Answers AN-1 Index of Applications IA-1 Index I-1 A-1 Preface his thirteenth edition of Calculus and Its Applications, and its Brief version, is written for either a one- or two-semester applied calculus course Although this edition reflects many revisions as requested by instructors across the country, the foundation and approach of the text has been preserved In addition, the level of rigor and flavor of the text remains the same Our goals for this revision reflect the original goals of the text which include: to begin calculus as soon as possible; to present calculus in an intuitive yet intellectually satisfying way; and to integrate the many applications of calculus to business, life sciences, and social sciences This proven approach, as outlined below, coupled with newly updated applications, the integration of tools to make the calculus more accessible to students, and a greatly enhanced MyMathLab course, make this thirteenth edition a highly effective resource for your applied calculus courses T The Series This text is part of a highly successful series consisting of three texts: Finite Mathematics and Its Applications, Calculus and Its Applications, and Brief Calculus and Its Applications All three titles are available for purchase as a printed text, an eBook within the MyMathLab online course, or both Topics Included The distinctive order of topics has proven over the years to be successful The presentation of topics makes it easier for students to learn, and more interesting because students see significant applications early in the course For instance, the derivative is explained geometrically before the analytic material on limits is presented To allow you to reach the applications in Chapter quickly, we present only the differentiation rules and the curve sketching needed Because most courses not afford enough time to cover all the topics in this text and because different schools have different goals for the course, we have been strategic with the placement and organization of topics To this end, the level of theoretical material may be adjusted to meet the needs of the students For example, Section 1.4 may be omitted entirely if the instructor does not wish to present the notion of limit beyond the required material that is contained in Section 1.3 In addition, sections considered optional are starred in the table of contents Prerequisites Because students often enter this course with a variety of prerequisite skills, Chapter is available to either cover in its entirety or as a source for remediation depending on vii viii PREFACE the pace of the course In addition to being covered in Chapter 0, some important topics, such as the laws of exponents, are reviewed again when they are used in a later chapter New to this edition, we have added a Prerequisite Skills Diagnostic Test prior to Chapter so students or instructors can assess weak areas The answers to the diagnostic test are provided in the student edition answer section along with references to areas where students can go for remediation Remediation is also available within MyMathLab through the newly created Getting Ready for Applied Calculus content at the start of select chapters New to This Edition This text has been refined and improved over the past twelve editions via the many instructor recommendations, student feedback, and years of author experience However, there are always improvements to be made in the clarity of the exposition, the relevance of the applications and the quality of exercise sets To this end, the authors have worked diligently to fine-tune the presentation of the topics, update the applications, and improve the gradation and thoroughness of the exercise sets throughout the text In addition, there are a few topics that the authors focused on to better enhance the learning experience for students r The Derivative (Chapter 1) In the previous edition, the derivative is introduced in Section 1.3 in an intuitive way, using examples of slopes of tangent lines and applied problems involving rates of change from Section 1.2 In the current edition, Section 1.3 incorporates an intuitive introduction to limits, as they arise from the computations of derivatives This approach to limits paves the way to the more detailed discussion on limits in Section 1.4 It offers the instructor the option of spending less time on limits (and therefore more time on the application) by not emphasizing or completely skipping Section 1.4 r The Integral (Chapter 6) Chapter has been significantly reworked As in the previous edition, we introduce the antiderivative in Section 6.1 However, we have simplified the presentation in Section 6.1 by opening with an example involving the velocity and position functions of a moving object This example motivates the introduction of the antiderivative or indefinite integral in a natural way Section 6.2 builds on the momentum from the examples of antiderivatives and introduces the definite integral using the formula b f (x) dx = F (b) − F (a), a where F is an antiderivative of f Several new examples are presented in Section 6.2 that illustrate the importance of the definite integral and the use of the antiderivative (net change in position, marginal revenue analysis, net increase in federal health expenditures) In Section 6.3, we introduce the concepts of Riemann sums and areas of regions under a graph, and prove the Fundamental Theorem of Calculus by showing that the Riemann sums converge to the definite integral Our new approach allows for an easier flow of the discussion of integration by moving directly from the indefinite integral to the definite integral, without the diversion into Riemann sums While the classical approach to the definite integral (as a limit of Riemann sums) has the concepts of limits and area as a driver, our approach emphasizes the applications as the driver for the definite integral More importantly, our new approach allows students to compute areas using basic geometric formulas (areas of rectangles and right triangles) and compare their results to those obtained by using the definite integral In contrast to the approach based on limits of Riemann sums, our approach provides a hands-on approach to areas and brings students closer to understanding the concept of Riemann sums and areas (See the Introduction and Example in Section 6.3.) PREFACE ix r Prerequisite Skills Diagnostic Test A quick assessment of the basic algebra skills students should already have mastered is provided prior to Chapter This can be a self-diagnostic tool for students, or instructors can use it to gain a sense of where review for the entire class may be helpful Answers are provided in the back of the student edition along with references to where students can go for remediation within the text r Now Try Exercises Students are now given Now Try exercises to encourage an immediate check of their understanding of a given example by solving a specific, odd-numbered exercise from the exercise sets r Additional Exercises and Updated Applications We have added many new exercises and have updated the real-world data appearing in the examples and exercises whenever possible r Chapter Summaries Each chapter ends with a summary that directs students to the important topics in the sections In addition, we have identified topics that may be challenging to students and presented several helpful examples Most notably, we have added examples that illustrate differentiation and integration rules, integration by parts, solving optimization problems, setting up equations arising from modeling, solving problems involving functions of more than one variable (Lagrange multiplier, second derivative test in two dimensions), and differential equations, to name just a few Effectively, the summaries contain more than one hundred additional, completely worked examples r Answers to Fundamental Concept Check Exercises We have added the answers to the Fundamental Concept Check Exercises so students can check their understanding of the main concepts in each chapter r Chapter Objectives The key learning objectives for each section of the text are enumerated in the back of the text These objectives will be especially helpful for instructors who need to verify that particular skills are covered in the text r Summary Endpapers A two-page spread at the back of the text lists key definitions, theorems and formulas from the course for an easy reference guide for students New Resources Outside the Text Annotated Instructor’s Edition New to this edition, an annotated instructor’s edition is available to qualified adopters of the text The AIE is a highly valuable resource for instructors with answers to the exercises on the same page as the exercise, whenever possible, making it easier to assign homework based on the skill level and interests of each class Teaching Tips are also provided in the AIE margins to highlight for new instructors the common pitfalls made by students Updates to MyMathLab (MML) Many improvements have been made to the overall functionality of MML since the previous edition However, beyond that, we have also invested greatly in increasing and improving the content specific to this text a Instructors now have more exercises to choose from in assigning homework b An extensive evaluation of individual exercises in MML has resulted in minor edits and refinements making for an even stronger connection between the exercises available in the text with the exercises available in MML c Interactive Figures have been developed specifically for this text as a way to provide students with a visual representation of the mathematics These interactive figures are integrated into the eText for student use, available in the instructor’s resources as a presentational tool, and are tied to specifically designed exercises in MyMathLab for targeted instruction and assessment The Interactive Figures run on the freely available Wolfram CDF Player Index of Applications Business and Economics Advertising, 55, 119, 121, 178, 187, 220, 225, 359, 391, 569 Amount of milk sold, 570 Annual rate of maintenance, 457 Annual sales, 277 Appreciation of assets, 239, 277 Area of a lot, 312, 317 Assembly lines, 25, 119, 329 Automobile population, 381–382 Automobile’s emergency flasher under warranty, 577, 578 Average annual income, 139 Average cost, 16, 204 Average price of MP3 players, 360 Average revenue, 204 Bakery making cookies, 584 Bank customers, 509–510 Bidding on a construction project, 556 Bonus plus taxes on taxes, 533 Break-even point, 28, 33, 65 Brokerage commissions, 6, 12 Bushels of wheat produced, 564, 565 Calls to a telephone switchboard, 581–582 Capital investment model, 499 Capital value of an asset, 451 Capitalized cost, 457 Car prices, 482 Catering company, 48 Company’s future earnings, 445 Comparing investments, 270, 289 Compound interest, 36–37, 38, 40, 129–130, 265–266, 271, 289, 290, 482, 512 Computer sales, 11, 121 Computing income tax, 97 Construction cost index, 472 Consumer price index, 140 Consumers’ surplus, 333–334, 337, 345 Continuous annuity, 500, 512 Continuous compound interest, 266–268, 269, 270, 271, 287, 289, 290, 337, 338, 345, 346 Cost analysis and production, 225 Cost functions, 15, 41–42, 48, 49, 54, 112, 139, 180, 188, 223, 247, 296–297, 300, 351 Cost of cleaning a pollutant, 20 Cost of fencing, 48 Cost of producing machine parts, 25 Cost of shipping and handling, 65 Cost, profit, revenue, 42–43, 48 Cost-benefit, 18, 20, 204 Daily business, 111 Daily output of a factory, 119, 178 Debt per capita, 71–72 Declining sales, 107–108 Decrease in sales in breakfast cereals, Demand and revenue, 186 Demand equation, 182–183, 188, 192–193, 209, 220, 225, 247, 277, 278, 342–343, 351, 360 Department store sales, 98, 111 Depreciation, 59, 239, 307 Distribution of revenue, 360 Earnings from a machine, 590 Economic lot size, 177, 179 Effect of an excise tax on sales price, 184–185 Effect of gas prices on miles driven, 381 Effect of stocks on total assets of a company, 212 Elasticity of cost, 277 Elasticity of demand, 273–276, 277, 290, 482 Estimating sales of toys, 112 Evans price adjustment model, 499 Expected assembly time, 569–570 Expected revenue from sales of monitors, 590 Factory accidents, 577 Farms in the U.S., 119, 139, 148 Federal Reserve banking, 550–551 Federal U.S minimum wage, 381 Fire insurance claims, 579 Fixed costs, 19, 20, 42, 58, 188 Fractional reserve banking, 550–551 Future value of an income stream, 334–335, 336, 337, 338 Gross Domestic Product, 272 Growth of money in a savings account, Hauling cost, 93 Home prices, 482 Impact of mad cow disease on beef export, 65 Income distribution, 589 Index-fund fees, 158 Insurance claims, 584 Interest rate, 187, 270–271, 338 Internal rate of return, 271, 523, 525 Inventory problems, 172–175, 176, 177, 178, 179, 192, 196, 345, 346 Investment analysis, 269 Isocost lines, 352 Isoquants, 216–217 Itemized deductions on a tax return, 66 Junk bond, 38 Lifetimes of manufactured products, 564, 569, 570, 572–573, 577, 578, 589 Machine break downs, 579, 585 Manufacturing, 94, 111, 505 Marginal costs, 58, 64, 65, 108–109, 117–118, 121, 129, 180–181, 211, 307, 328, 329, 331, 345 Marginal productivity of labor and capital, 357–358, 360 Marginal profit, 111, 121, 211, 324, 329, 330 Marginal propensity to consume, 533 Marginal rate of substitution, 216–217 Marginal revenue, 111, 208, 305, 340, 345 Maximizing production, 371–372 Maximizing profits, 183–185, 186, 196, 367, 375 Maximizing revenue, 46–47, 178, 182, 185, 186, 187, 196 Maximizing total orchard yield, 196 Minimizing cost, 166–167, 169, 170, 171, 178, 201, 375 Minimizing marginal cost, 186 Minimizing space in a firm, 374 Monopolist’s demand, 65 Monthly interest rate, 524 Mortgage payment, 307, 359 Multiplier effect, 529–530, 533 Net increase in an investment, 307 New coke ovens, 492 Number of subway passengers, 129 Oil consumption, 139, 325–326, 330 Optimal airline fares, 188 Optimal amount of labor, 375 Ordering furniture and receiving delivery, 577, 578–579 Paying off a car loan, 478–479 People arriving at a supermarket, 585 Perpetuity, 533, 551 Predicting profits, 109–110 Predicting sales, 111 Present value of an income stream, 338, 442–443, 445, 446, 455, 457 Present value, 270, 289, 351, 391 Price affects sales, 65, 121 Price discrimination, 363 Price of a company’s stock, 121 Price of crude oil, 71 Price of gasoline, 65 Price of wheat, 277 Price setting, 186–187 Probability of gasoline sales, 589 Producers’ surplus, 337 Production, 43–44, 185, 299, 375 Production functions, 117, 349–350, 351, 359–360, 376 Profit functions, 21, 28, 48, 186, 187, 211, 300, 367 IA-1 IA-2 Index of Applications Profitability of franchise operations, 561 Property tax, 457 Quality control, 557, 558, 582, 583, 585 Quit ratio, 65 Rate of change of price, 220 Rate of change of taxes, 223 Rate of net investment, 464 Rate of output of a coal mine, 129 Real estate investment, 269, 270 Relationship between price and sales, 471 Retirement accounts, 477–478, 481–482, 508 Revenue from sales, 44–45, 49,55, 97 Revenue functions, 111, 181, 188, 224, 247, 347, 367 Sale of T-shirts, 65 Sales decay curve, 261–262 Sales of computers, 111 Savings account, 269, 270, 337, 390, 464, 481, 482, 483, 495, 500 Selling insurance, 569, 571 Service contract, 589, 590 Successful restaurants, 569 Tax and homeowner exemption, 351–352 Taxes, profit and revenue, 187 Technology stock, 269 Time between arrivals at a tollgate,577 Time between buses, 570 Time required serving a bank customer,577 Time to complete a job, 570 Tolerance limits, 590 Two competing companies, 585 U.S consumption of iron ore, 300 U.S electrical energy production, 194, 195 U.S federal debt, 71, 112, 134 U.S natural gas production, 300 U.S Treasury bill, 119 Value of an investment, 272–273, 277, 287–288 Variable costs, 19, 20, 42 Waiting time, 564 Weekly pay and volume of sales, 65 Weekly sales and price, 218 Zero coupon bond, 38 Health and Life Sciences Acetylcholine and heart muscle contraction, 11 Age of a cell, 562, 567 Allometric equation, 211, 250 Anesthetics, 224 Babies born each day, 585 Bacteria growth, 2, 140, 257–258, 262, 265, 286, 287, 289, 290, 483, 499 Bacterial infection, 584 Basal metabolic rate, 407–408 Birth rate, 512 Blood flow through the brain, 139 Blood pressure, 407 Blood test probabilities, 589 Body mass index, 205 Body surface area, 360 Carbon dating, 261, 264, 289 Carbon dioxide diffusion in lungs, 464 Carbon monoxide levels, 55, 211 Cardiac output, 434, 440 Cell population, 558–559, 560, 565 Changes in body weight, 163 Concentration of a drug in the bloodstream, 12, 133, 148, 243, 286 Concentration of glucose in the bloodstream, 285 Deaths due to a disease, 580–581 Decay constant, 259–260 Decay of penicillin in the bloodstream, 263 Decay of sulfate in the bloodstream, 264 Deforestation and fuel wood, 329–330 Determining the time of death, 482 Dialysis and creatinine clearance, 483 Doctors adopting a new drug,285 Drug constant, 264 Drug dosage, 533 Drug injected into a patient, 345 Elimination of a drug, 121, 500, 530–531, 533 Enzyme kinetics, 19 EPA fine for pollution, 14 Epidemics, 250, 282–284 Escherichia coli bacteria, 262 Exponential decay, 259 Exponential growth, 257, 263 Eye pupil size and light intensity,205 Fish population in a lake or pond, 282 Flu epidemic, 300, 462, 465 Food chain, 24 Glucose elimination, 285 Gompertz growth curve, 239, 472 Growth constant, 258 Growth of a plant, 491 Growth of a tumor, 220, 239–240 Growth of cells, 262 Growth of fruit flies, 258–259 Half-life, 259–260, 264, 265, 289, 337 Health expenditures, 121, 255,305, 381 Height of a child, 129 Height of a plant, 163, 239, 240, 290 Herring gull population, 289 Insect population, 262 Intravenous infusion of glucose, 280–281 Intravenous injection, 65 Iodine level in dairy products, 260 Logistic growth, 281–282 Lung physiology, 284 Lungs and trachea clearing, 175–176 Morphine infusion, 484 Net primary production of nutrients, 16 Oil spill in ocean, 225 Oxygen in a lake, 178 Paramecium growth, 464 Patient’s temperature, 139 Pollution, 14, 17–18, 139 Population genetics, 496–498, 500 Population growth, 262, 263, 289, 337, 463, 493–494 Population of fish, 464, 472, 494–495, 499–500 Predator-prey model, 404–405 Probability of an infection, 585 Protozoa in water samples,579, 582 Relief time for arthritic patients, 590 Survival probabilities, 577 Therapeutic level of a drug, 484 Time of birth, 578 Tooth cavity, 585 Transmural pressure in lungs, 140 Volume of air in a person’s lungs, 416–417 Waiting time in an emergency room, 584 Water level, 195, 307 Weight and surface area of a horse, 225 White blood cell count, 590–591 Physical Science Accidents occurring at an intersection, 558, 585 Analysis of a moving particle, 119 Atmospheric pressure, 255, 289 Autocatalytic reaction, 499 Average speed, 120 Average temperature, 337, 408 Average velocity, 120, 204, 336 Breaking weight, 578 Car travel, 120 Carbon dioxide level in a room, 500 Changes in temperature, 147 Chemical reactions, 499, 512 Decay of radioactive iodine, Displacement versus distance traveled, 324–325, 330 Distance traveled, 439 Diver’s ascent, 65 Elevation and temperature in a creek, 382 Evaporation, 505 Free fall, 299 Gravity, 499 Heat diffusion, 300, 329 Height of a building, 410 Height of a helicopter, 120, 129, 329 Height of an object in the air, 45, 49, 120, 412 Length of a metal flange, 575–576 Level of water from melting snow, 147 Maximum height reached by a ball, 164–165 Motion of an object, 119, 122 Motorcyclist driving over a ramp, 221 Movement of solutes through a cell membrane, 499 Net change in position, 305, 307 Newton’s law of cooling, 461, 464, 480–481 Oil in a storage tank, 58–59, 68 One-compartment mixing process, 496 Position of a ball, 115–116, 127, 299 Position of a rocket from its velocity, 292 Radioactive decay, 263, 264, 289–290, 307, 483 Rate of change of volume of liquid, 119 Rate of decomposition, 472 Index of Applications Stopping distance of a car, 33, 40 Tap water temperature, 400–401 Temperature in a city, 62 Temperature of a liquid, 118, 121 Temperature of a rod, 290, 482 Total distance traveled by a bouncing ball, 533 Velocity and acceleration, 114–115, 121–122, 195 Velocity and distance of a rocket, 330 Velocity of a baseball, 20 Velocity of a car, 329 Velocity of a parachutist, 140, 239, 491 Velocity of a rock, 345 Velocity of a skydiver, 278, 307, 464 Volcano eruption, 446 Volume and pressure of a gas, 220, 360 Water flow, 346 Width of a river, 412 Wind velocity, 244 Social Sciences Amount of information learned, 472 Demographic model, 444–445, 446 Diffusion of information by mass media, 279–280, 286 Distribution of IQs, 435–436 Ebbinghaus model for forgetting, 285 Expected reading time, 570 Judgment time, 122 Learning curve, 279 Model in psychology, 471 Population density of a city, 446 Population near New York City, 196 Population of Mexico City, 263 Population of Texas, 289 Population of the Pacific Northwest, 264 Population of the United States, 113–114, 158, 204 Population with emigration, 307, 479–480 Social diffusion, 499 Spread of news, 285, 286, 464 Students using a library, 568–569 U.S Supreme Court vacancies, 577 Urban population, 66 War fever, 499 World’s population, 263, 333 General Interest Airplane observer, 220 Amount of milk in a container, 578 Appreciation of art work, 267–268, 269 Area of a property, 439 Area of the vertical cross section of a river, 439 Baseball diamond, 220–221 Beer consumption, 360 Buffon needle problem, 590 Car accident related deaths in the U.S., 379 Citrus growers’ response to predicted freezing weather, 558 Coffee consumption in U.S., 179 Coin tossing, 582, 585 Consumption of ice cream in the U.S., 204 Conversion from Celsius to Fahrenheit, 6, 65 Conversion of men’s hat sizes, 25 Cost of car rentals, 20 Crime rate, 391 Daylight hours, 134, 401, 408 Design of a wind shelter, 171 Diameter of a bolt, 578 Dimensions of a Norman window, 170 Distribution of typos, 584 Drying a porous material, 499 Exam grades, 552–554 Fencing, 48, 169, 170, 176 Graduating seniors, 388–389 Heat loss of a building, 348–349, 357, 363–364 Height requirements by a police department, 590 IA-3 Insect repellent, 247 King Arthur’s Round Table, 264 Ladder position, 220 Maximizing area, 165–166, 169, 170, 176, 178, 186, 196, 247, 374–375, 391 Maximizing volume, 167–168, 169, 171, 178, 179, 191–192, 196, 204, 367, 374, 376, 403 Medical expense, 20 Minimizing area, 170, 178, 179, 196, 223, 367, 376 Minimizing the sum of the distances of two towns to a highway, 171 Minimizing time, 196 Minimizing volume, 169, 170, 391 Modes of transportation, 360 Number of cars at a tollgate, 584 Paper jam in copy machines, 585 Phone calls coming into a switchboard, 558 Position of a toy rocket, 120 Probability of accidents, 471 Rate of litter accumulation, 498, 500 Right to drill, 20 Rolling dice, 591 Roulette, 555 Saline solution, 307 SAT score distribution, 578 Scores on an entrance exam, 590 Selecting a number at random, 555, 566–567 Spinning a spinner, 558 Target shooting, 585 Throwing a skewed die, 586 Time between phone calls, 573 Time of a commute, 578 Time of the fourth ice age, 264 Use of books at a library, 225 Volume of a mothball, 472 Walking, 129 World rate of water use, 346 This page intentionally left blank Index A Absolute maximum point, 133 Absolute maximum values, 133 Absolute minimum values, 133 Absolute-value functions, 18, 52 Acceleration defined, 115, 127 function, 115 Addition, rational functions, 22 Algebraic expressions, 35 Angles, 392–395, 413 Antiderivatives cost function, 296–297 defined, 292 determination of, 432 differential equations, solving for, 295–296 exponential function, 292–293 power function, 292 of same function, 293 theorems, 293 Antidifferentiation defined, 292 position of rocket from its velocity, 292 as reversing derivative formula, 297 Applications See Index of Applications Applied problems business, 42–43 functions and graphs, 43–46 geometric, 40–42 slope representation in, 56 translating, 46 Approximating solutions Euler’s method for, 501 Newton–Raphson algorithm, 522–523 to polygonal paths, 503 Approximation of definite integrals See also Definite integrals applications of, 442–446 cardiac output measurement example, 434–435 error analysis, 437–438 geometric output, 436–437 midpoint rule, 433 reasons for, 432–433 Simpson’s rule, 435–436 trapezoidal rule, 433–434 Approximations accuracy of, 517–518 change in function, 116–118 linear, 514 with Newton–Raphson, 521–522 production function, 117 sequence to r, 521 Taylor polynomials, 515–516 zeros of polynomials, 522 Area between two curves area of the region formula, 322 defined, 320 and points of intersection, 322–323 Area bounded by curves, 319–320 Area function approximation, 327 Theorem I, 326–327 Area under graphs cubic, 310 defined, 308 definite integral and, 308–317 estimation, 311 of functions, 310 illustrated, 308–309 parabola, 308 of rate of change, 323 Riemann sum approximation, 311–315 Theorem I, 309–310 Theorem II, 314–315 Areas circle formula, 168 defined, 41 of ellipse, 430–431 improper integral, 448 maximizing, 165–166 under normal curve, 449 probabilities displayed as, 556 rectangle formula, 168 Riemann sum approximation, 312–313 signed, 318 under sine curve, 404 surface, 41–42 in theory of calculus, 326–327 in xy-plane, 318–331, 342 Asymptotes, 136–137, 161 Autonomous differential equations See also Differential equations Average cost, 201 Average rate of change, 112–113, 127 Average value of functions, 332 Average velocity, 115 B Ball height of, 45 maximum height reached by, 164–165 position of, 115–116 Base, 226 Beta probability density, 561–562 Break-event points, 28 bx , 238 C Carbon dating, 261 Carrying capacity, 493 Carrying costs, 172 Chain rule applying, 208, 236 defined, 207, 222 for exponential functions, 236 form, 206 general power rule as, 207 verification of, 209–210 Change function, approximating, 116–118 of limits rule, 430 Circles area, 168 circumference, 168 Clockwise angles, 394 Cobb-Douglas production function, 349 Common logarithms, 242 Comparison test applying, 538 defined, 537, 548 Compliance, total, 284 Composite function, 206 Composition of function application, 24 defined, 23, 206 Compound interest compounded continuously, 266 defined, 36 effect of increased periods, 266 exponents in, 36–38 formulas, 36–37 interest period, 36 interest rate, 37 monthly compound, 38 present value and, 268 principal amount, 36 rate of growth, 267 zero coupon bond, 38 Concavity checking, 160 defined, 135 in extreme point determination, 152–154 of solution curves, 489 Cones, volume of, 336 Constant functions, 13–14, 52, 74 Constant relative rate of change, 273 Constant solutions See also Solutions of differential equations defined, 460 differential equations, 488–489 graphing and, 488–489 Constant-multiple rule See also Differentiation defined, 98 proof of, 101 Constrained optimization problem, 368 Constraint equation, 191 Consumer surplus defined, 334 definite integrals in, 333–334 demand curve, 333 Continuity, 92–98 Continuous functions, 125 Continuous random variables See also Probability; Random variables defined, 558 probability density, 559–562, 563–564 random selection, 562–563 I-1 I-2 Index Convergent improper integrals, 448–449 Convergent series defined, 528, 534, 548 determining, 536–538 Cosecant function, 409 Cosine See also Sine alternative definition of, 397 calculation of, 396 defined, 396 derivatives with, 402 differentiation of, 401–405, 414 graph of, 399 informal justification of differentiation rules for, 405–409 in maximizing a volume, 403 in predators-and-prey model, 404–405 properties of, 397–398 rules, 402 in solving right triangle, 397 values of, 396, 397 Cost functions antiderivatives, 296–297 business and economic derivative applications, 180–181 graph of, 43 inventory problem, 174 linear, 15 manufacturing plant, 94 Cost-benefit model, 17–18 Costs average rate of change, 201 business problems, 42 fixed, 15, 58 functions and graphs, 43–44 geometric problems, 41 hauling, 93 marginal, 58, 108–109, 117–118, 144 minimizing, 166–167 variable, 15 Cotangent function, 409 Counterclockwise angles, 394 Critical numbers, 150 Critical points defined, 150 in finding extreme points, 164 no, functions with, 160 Critical values defined, 150 finding, 153 Cubes formula, 168 Cubic, area under graph of, 310 Cumulative distribution function, 561 Curve sketching defined, 149 finding inflection points, 162 finding relative extreme points, 161–162 function properties and, 162 technique summary, 161 Curves area between, 320–323 area bounded by, 319–320 demand, 182, 218, 333 level, 349–351 logistic, 493 normal, 573, 574 slope of, 66–73 solution, 486, 489 tangent lines to, 67 Cylinders, 41, 167–168 D Decay constant defined, 259 half-life and, 259–260 Decimal expansion, 529 Declining sales, 107–108 Decreasing, slope, 134–135 Definite integrals antidifferentiation and, 292–300 in approximating sums, 331 approximation of, 432–441, 454–455 area under graphs and, 308–317 in average value, 332–333 computing, 301–302, 306 defined, 301 evaluation of, 429–432, 453–454 of exponentials, 302–303 of functions, 300–308 in future value, 334–335 integration of parts in, 431 limits of integration, 301 marginal revenue analysis, 305 net change in position, 305 net change of functions and, 300–308 by parts, 431 of piecewise-defined function, 304 properties of, 302 properties of, using, 303–304 in solids of revolution, 335–336 with substitution, 429–430 Taylor series, 543 Degrees, converting to radians, 393–394 Demand curves in consumer surplus, 333 illustrated, 182, 218 Demand equation defined, 182 setting up, 182–183 Demographic model illustrated, 444 Dependent variables, Depreciation of assets example, 59 Derivatives See also Antiderivatives computing from definition, 87 computing from limit definition, 86–87 conditions on, 142 of constant function, 74 with cosine function, 402 defined, 73 evaluation at one point, 75–76 evaluation of, 106 of exponential, 231–232 first, 141 forming from functions, 77–78 of function with radical, 88 as fundamental calculus tool, 57 geometric meaning of, 74–78 involving ex , 233 of ln x, 244–247 limit definition of, 86–88 limits and, 73–82 of linear function, 73 notation for, 105–106 partial, 353–361 product of two functions, 197 as rate of change, 106–107, 112–122 of rational function, 87–88 recognizing limits as, 88 related rates, 217–218 rules, 141–149 secant-line calculation of, 78–79 second, 105, 106, 126, 142 with sine function, 402 in solving tangent problems, 77 symbols, 207 tangent functions, 410–411 tests, 149–158 of x2/x3 , 75 of y , 214–215 Differences algebraic operation, 52 of cubes, finding, 31 Differentiability, 92–98 Differentiable functions, 125 Differential equations applications of, 492–501, 509–510 autonomous, 484, 485–486 concavity of solution curves, 489 constant solutions, 460, 488–489 defined, 237, 458 first-order, 458, 459, 473–477, 507–508 first-order, applications, 477–484 geometric modeling of, 461–463 graphing solutions, 484–492, 508–509 initial conditions, 459 with initial values, 237 initial-value problems, 459–460 logistic, 492–494 modeling with, 460–461 Newton’s law of cooling example, 461 numerical solutions of, 501–506, 510–511 in population genetics, 496–498 qualitative theory of, 484 second-order, 459 separation of variables, 466–473 setting up, 492–494 slope fields, 461–463 solution curve slope, 486 solutions of, 458–465, 506 solving, 237, 295–296, 459 verifying solutions, 458–459 working with, 258–259 Differential rules chain rule, 206–212 general power rule, 206–212 multiple, using, 200–201 product, 197–199 quotient, 199–200 Differentiation constant-multiple rule, 98–99, 101 of cosine, 401–405, 414 defined, 73 of exponential functions, 235–240 exponentials, 233–234 general power rule, 98, 100–101 implicit, 212–221 logarithmic, 249 natural logarithm function and, 248–249 product rule, 198–199 with respect to specific variables, 104 rules for, 98–103 Index rules for sine and cosine, 405–409 of sine, 401–405, 414 sum rule, 98, 99–100, 102 Taylor series, 541–542 techniques, 197–225 Diffusion of information defined, 279 example, 280 graph, 280 by mass media, 279–280 Discontinuous functions defined, 93 illustrated, 94 occurrence of, 94 Discrete random variables See also Probability; Random variables decision-making based on expected value, 556 expected values, 553–554 expected values and variance, 555 outcomes, 554 relative frequencies, 553–554 summary, 586 variance, 555 Distribution, 435–436 Divergent improper integrals, 448–449 Divergent infinite series See also Infinite series defined, 528, 548 determining, 536 Division, rational functions, 23 Domains of functions defined, finding, rational functions, 17 restricted, specifying, 6–7 Double integrals defined, 382 evaluation, 383–384 region, 382, 383, 385 solid boundary, 383 value calculation, 383 volume using, 385 E Economics cost functions, 180–181 derivatives applied to, 180–188 functions of several variables in, 349–350 laws of, 372 marginal concept, 108–110 multiplier effect in, 529–530 natural logarithm function in, 271–278 production functions in, 349–350 profit functions, 183–185 revenue functions, 181–183 setting production levels, 185 E k x , 238 Elasticity of demand defined, 273, 274 relative rate of change comparison, 274 Elimination of a drug example, 530–531 Ellipses, area of, 430–431 End point extremum, 176 Endpoint extreme value, 133 Epidemic model, 282–284 Error analysis approximation, 437–438 theorem, 437 Euler’s method, 501–503 Ex applying laws of exponents with, 232 defined, 230 derivatives involving, 233 exponential function, 230–235 fundamental properties of, 232 Taylor polynomial approximation of, 515 Expected value decision-making based on, 556 defined, 554–555, 566 defining, 579 interpretation of, 566 variance and, 555 Exponential decay defined, 226, 259 half-life and, 259–260 rate of, 259 of sales, 261–262 time constant in, 262 Exponential density functions, 571 Exponential growth, 226, 257–258 Exponential random variables, 571, 587 Exponential rule, 294 Exponents algebraic manipulation of, 34 antiderivative, 292–293 base, 226 b x , 238 chain rule for, 236 in compound interest, 36–38 continuous, 268 defined, 226 definite integrals of, 302–303 derivative of, 231–232 differentiation of, 233–234, 235–240, 252 e k x , 238 e x , 230–235 graph of, 233 laws of, 34–35, 227–228 product of x and, 426 simplifying, 241–242 solving equations with, 228–229, 242 substitution with, 420–421, 422 tangent line to graph, 232 Extreme points, 132–133, 152–154, 164 F Factoring, 28–30, 36, 85 Finite sums, 532 First derivatives, graphing, 143 First-derivative rule defined, 141, 189–190 graphing using properties of the derivative, 142 illustrated, 141 First-derivative test applying, 151–152 defined, 150, 190 functions of two variables, 362 use selection, 156 using, 160–161 First-order differential equations See also Differential equations I-3 applications of, 477–484, 508 defined, 459, 473, 507 initial-value problem, 475–476 integrating factor, 473 linear, 473–477 Newton’s law of cooling example, 480–481 solving, 474–475 steps to solving, 475 summary, 507–508 typical solutions to, 474, 475 Fixed costs defined, 15 slope of lines, 58 Fractional powers factoring, 36 substitution with, 422 Fractions, properties of, 21 Functions of several variables defined, 347 double integrals, 382–386 in economics, 349–350 first-derivative test, 362 heat loss, 363–364 Lagrange multipliers, 368–376 level curves, 350–351 locating minimum values, 362 maxima of, 361–367, 387–388 method of least squares, 376–382 minima of, 361–367, 387–388 price discrimination, 363 second-derivative test, 365–366 three variables example, 347 two variables example, 347–348 Fundamental theorem of calculus, 314 Future value of income stream, 334–335 G Gene frequency, 497 General power rule as chain rule, 207 defined, 98 differentiation of radicals, 100 implicit differentiation and, 214–215 Geometric distribution, 583 Geometric formulas, 168 Geometric interpretation, of partial derivatives, 355–356 Geometric problems, 41–42 Geometric random variables See also Random variables defined, 583 trial results, 582–583 Geometric series See also Infinite series defined, 528 rational number of, 529 sigma notation and, 531–532 Graph of the equation, 10 Graphing constant solutions and, 488–489 derivatives, 145 differential equations, 484–492 with first and second derivatives, 143 functions, 10–11 intervals, with properties of the derivative, 142 trigonometric functions, 399 ln x, 246 I-4 Index Graphing Calculator antidifferentiation, 297 approximating integrals, 438 approximating slope of a graph at a point, 69–70 area between two curves, 327 computing definite integral, 306 computing zeros of functions, 32 Euler’s method, 503–504 evaluating functions, 19 evaluating partial derivatives, 359 finite sums, 532 graphing derivatives, 145 graphing functions, 10–11 graphing trigonometric functions, 399 improper integrals, 450 least-squares method, 380 maximum/minimum value approximation, 46–47 Newton–Raphson algorithm, 524 numerical derivatives and tangent lines, 79 piecewise-defined functions, 96 Poisson probabilities, 583–584 Riemann sums, 315 scientific notation, 39 slope fields, 463 solving equations and intersection of graphs, 229 Taylor polynomials, 518 Graphs approaching asymptotes, 136–137 computing limits with, 83 connections between, 143–144 cosine, 399 describing, 137 intercepts of, 136 intersection of, 27–28 points moving along, 217 sine, 398–399 with undefined points, 136 Graphs of functions in applications, 40–50 with asymptotes, 161 costs, 43–44 curves that are not, defined, 7, 50 describing, 131–141 exponential, 227–228, 233 linear, 51 by plotting points, 7–8 quadratic, 51 reading, with restricted domain, 8–9 revenue, 44 tangent, 411 vertical line test, 9–10 Growth constant, 257–258 H Half-life, 259–260 Harmonic series, 528 Heat-loss function in architectural design, 348–349 functions of several variables, 363–364 partial derivatives, 348–349 I Implicit differentiation See also Differentiation defined, 212–213 finding slope with, 213–214 general power rule and, 214–215 general procedure for, 216 graph illustration, 213 related rates and, 217–218 theorem, 215 using, 215–216 Improper integrals area defined by, 448 convergent, 448 defined, 446 divergent, 448–449 upper limit, 447 Incorporating Technology, 24, 63, 110, 203, 219, 234, 576 Graphing Calculator, 10–11, 19, 32, 39, 46–47, 69–70, 79, 90, 96, 145, 229, 297, 306, 315, 327, 359, 380, 399, 438, 450, 463, 503–504, 518, 524, 532, 583–584 Indefinite integrals, 293 Index of summation, 531 Inequalities, 3–5 Infinite series comparison test, 548 convergent, 528, 534, 535, 548 defined, 527, 547 divergent, 528, 535–536, 548 geometric, 528–529 geometric picture, 535 harmonic, 528 integral test, 536–538, 548 multiplier effect in economics, 529–530 with positive terms, 534–540, 548 representation by rectangles, 535 sigma notation, 531–532 sum association, 527 term-by-term comparison, 537 Infinity, 4, 89–90 Inflection points defined, 135, 190 illustrated, 135 location of, 144, 154–156 Initial conditions, 459 Initial side, angle, 394, 413 Initial-value problems, 459, 468–470, 475–476 defined, 459 Input, business problems, 42 Instantaneous rate of change average rate of change comparison, 113 defined, 113, 127 Integral formulas, 297 Integral sign, 293 Integral test, 536–538, 548 in divergent determination, 536 Integrals applications of, 442–446 approximating, 432–441 in approximating sums, 331 basic properties of, 295 definite, 300–339 double, 382–386 evaluation of, 331 exponential rule, 294 improper, 446–451, 455–456 indefinite, 293–294 log rule, 295 of piecewise-defined function, 304 power rule, 294 of rates of change, 304 of rational function, 422–423 of sine functions, 403 Taylor polynomials in approximating, 516 using properties of, 303–304 Integrands, 294 Integrating factors, 473 Integration approximation of definite integrals, 432–441 change of limits rule, 430 of cosine, 401–405, 414 defined, 418 evaluation of definite integrals, 429–432 formula derivation, 419 formulas, 419 of sine, 401–405, 414 Taylor series, 541–542 techniques of, 418–457 Integration by parts applying twice, 427 defined, 425–426 formula, 425 logarithmic function, 427–428 product of power of x and logarithm, 427 product of x and exponential function, 426 product of x and trigonometric function, 426 Integration by substitution defined, 419 definite integrals with, 429–430 with exponential, 420–421, 422 formula, 419 with fractional powers, 422 integral of rational function, 422–423 with logarithm, 421–422 method of least squares, 421 with radical, 421 with trigonometric functions, 423 Integration of parts in definite integrals, 431 in present value of income stream, 443 Interest period, 36 Interest rate, 37 Internal rate of return example, 523–524 Intervals, Intrinsic rate of growth, 494 Inventory control See also Optimization problems carrying costs and, 172 cost function, 174 economic lot size and, 175 economic order quantity and, 173 ordering costs and, 172 problem, 172–173 Irrational numbers, Isoquants marginal rate of substitution and, 216–217 production, 216 J Junk bonds, 38 Index L Lagrange multipliers defined, 368 generalization to functions, 372 marginal heat loss, 374 in maximizing production, 371–372 method of, 368 setting partial derivatives and, 368–369 theorem, 368–369 in three variables, 373–374 Laws of economics, 372 Laws of exponents applying with ex , 232 defined, 34 operations with, 35–36 using, 227–228 Learning curve, 279 Least-squares error See also Method of least squares defined, 377 Least-squares line, 377 Length, 40 Level curves defined, 349 illustrated, 349 of production function, 351 Limit formula for e, 268 Limit theorems, 84–85, 96, 101–102, 202, 210 Limits definition of derivative, 86–88 of functions, 83 infinity and, 89–90 of polynomial function, 85 properties of, 84 of rational function, 85 recognizing as derivative, 88 secant-line calculation of, 78–79 as x approaches infinity, 89 Linear approximations, 514 Linear functions constant, 13–14 cost, 14 defined, 13 derivative of, 73 graph of, 51 intercept determination, 15–16 sketching, 14, 16 slope of line and, 56 x-intercept, 15 y-intercept, 15 Lines equation given point and slope, 61 equation through two points, 61 fitting to data, 376–377 least-squares, 377 nonvertical, equations of, 57 parallel, 60 perpendicular, 60 secant, 78–79 sketching given point and slope, 60 slope of, 57–66 tangent, 67 ln x See Natural logarithm function Log rule, 295 Logarithmic derivative, 271 Logarithmic differentiation, 249 Logarithmic function, 427–428 Logarithms to base 2, 242 common logarithms, 243 differentiating after simplifying, 248–249 natural, 241–243 product of power of x and, 427 simplifying expressions with, 248 substitution with, 421–422 using properties of, 248 Logistic curves, 493 Logistic differential equations defined, 509 xy-graph for, 493 Logistic growth curve, 281–282 equation of, 281 M Manufacturing cost, 94 Marginal cost analysis, 180–181 approximation, 117–118 curve, 144, 201 defined, 58, 108 functions, 181, 201 of producing units, 108 slope of lines, 58 Marginal productivity, 357–358, 372 Marginal profit, 324 Marginal propensity to consume (MPC), 530 Marginal rate of substitution defined, 217 isoquants and, 216–217 Marginal revenue analysis, 305 curve, 144 defined, 109 function, 182–183 time rate of change and, 208 Mathematical models, 460–461 Maximum values approximating, 46–47 defined, 133 Maximum/minimum points defined, 361 first-derivative test, 362 illustrated, 361 second-derivative test, 365–366 Method of least squares defined, 376 fitting a straight line to data, 376–377 least-squares error, 377–378 least-squares line, 377 Midpoint rule See also Approximation of definite integrals defined, 433, 454 error from, 435 geometric interpretation, 436–437 Minimum values approximating, 46–47 defined, 133 locating, 362 Minus infinity symbol, I-5 Modeling with differential equations, 460–461 with Poisson distribution, 582 Multiplication, rational functions, 23 Multiplier effect, 529–530 Mutations, 498 N Natural logarithm function analyzing functions involving, 245–246 computing derivative of, 244 defined, 241 derivative of, 244–247, 253 in economics, 271–278 elasticity of demand and, 273–276 graphing, 246 as inverse of exponential function, 241 as logarithm to base e, 242 percentage rate of change and, 288 properties of, 247–250, 253 relative rates of change and, 271–273, 287 solving equations with, 242 use in calculus, 243 verification of, 247–248 Net change area under rate of change and, 323 defined, 300 definite integral and, 300–308 increase in federal health expenditures, 305 marginal revenue analysis, 305 in position, 305 vertical shift and, 301 Net displacement, 325 Newton–Raphson algorithm approximating solutions, 522–523 approximating zeros of polynomials, 522 approximation using, 521–522 defined, 520, 547 illustrated, 521 iterating, 522 Newton’s law of cooling first-order differential equations, 480–481 solutions to differential equations, 461 Nondifferentiable, 86, 125 Nondifferentiable functions defined, 92 illustrated, 93 Normal curves defined, 573 standard, 574 Normal density functions defined, 573 illustrated, 574 random variables with, 574 Normal random variables See also Random variables defined, 573 standard, 574–575 Normally distributed values, 573 Number line, intervals on, Numbers critical, 150 irrational, rational, 529 real, 3–13 I-6 Index O Objective equation, 191 Operations on constant functions, 52 on functions, 21–22 with laws of exponents, 35–36 on power functions, 53 on Taylor series, 541–543 Optimization problems constrained, 368 constraint equation, 191 defined, 164 end point extremum, 176 geometric formulas, 168 inventory control, 172–175 objective equation, 191 Ordering costs, 172 Outcomes defined, 554 representation, 579 Output, 42 P Parabolas, approximation by, 437 Parallel lines, 60–61 Partial derivatives See also Derivatives bracket use in computing, 355 computing, 353–354 defined, 353, 387 differentiation rules and, 354–355 evaluating, 355, 359 geometric interpretation of, 355–356 heat-loss function, 357 of higher order, 358 interpreting, 356–357 in marginal productivity of capital, 357–358 rates of change and, 356–359 setting in method of Lagrange multipliers, 368–369 Partial sums alternation, 527–528 defined, 527 growth of, 527 Percentage rate of change, 288 Perimeter, 40, 168 Perpendicular lines, slope of, 60 Piecewise-defined functions, 96 Points break-even, 28 continuity at, 95–96 extreme, 132–133 fitting straight lines to, 376–377 inflection, 135 of intersection, area between two curves and, 322–323 line equation given slope and, 61 line sketching given slope and, 60 maximum, 361 minimum, 361 moving along graph, 217 reflection of, 240 slope of curve at, 66 two, equation through, 61 Poiseuille’s law of fluid flow, 284 Poisson distribution See also Random variables computing probabilities with, 581–582 defined, 580 modeling with, 582 Poisson random variables, 580, 588 Polygonal paths, 501 approximating solutions to, 503 Polynomial functions, 17 limit of, 85 Polynomials approximating zeros of, 522 as function, 513 quadratic, 29–30 Taylor, 513–520 Population, 496 gene frequency in, 497 intrinsic rate of growth for, 494 logistic model for change, 493 Power functions antiderivative, 292 defined, 18, 52 performing operations on, 53 Power rule applying, 74 defined, 74 general, 98, 100–101 integral, 294 verification of, 79, 249 Power series convergence of, 544 defined, 540, 548 domain of, 544 radius of convergence, 544 Predators-and-prey model, 404–405 Present value concept, 443 of continuous stream of income, 443 defined, 268 of income stream, 442–443 of total income, 443 Price discrimination, 363 Principal amount, 36 Probability area of rectangle and, 560 calculus and, 552–591 continuous random variables and, 558–565 discrete random variables and, 552–558 displayed as area, 556 expected value and, 554–556, 566–571 exponential/normal random variables and, 571–579 of occurrence, 579 Poisson/geometric random variables and, 579–586 relative frequencies, 554 with standard normal random variable, 574–575 table, 556 Probability density beta, 561–562 defined, 559 determining, 560–561 function, 559 uniform, 567 working with, 563–564 Product, algebraic operation, 52 Product of x exponential function and, 426 logarithm and, 427 trigonometric function and, 426 Product rule applying, 198 defined, 197, 221 differentiating and simplifying, 198–199 verification of, 202 verifying, 198 Production levels, setting, 185 maximizing, 371–372 Production functions approximating, 117 Cobb-Douglas, 349 in economics, 349–350 level curves of, 351 Production isoquant, 217 Profit break-event points and, 28 business problems, 42–43 functions, 183–185 marginal, 109 maximizing, 183–184 predicting, 109–110 Proofs constant-multiple rule, 101 sum rule, 102 Proportional positive/negative and, 461 rate of change, 492 Q Quadratic formula defined, 26 derivation of, 31 using, 26–27 x -intercept computation, 159 Quadratic functions defined, 17 in economics, 16–17 finding zeros of, 27 graph of, 51 Quadratic polynomials, 29 Qualitative theory of differential equations, 484 Quotient rule defined, 199, 221 simplifying after, 199–200 using, 199 verification of, 202 Quotients algebraic operation, 52 rational functions, 23 R Radians converting degrees to, 393–394 defined, 392 measure of angles, 392–395 measurement example, 394 relations, 393 Radicals differentiating, 100 functions with, derivatives of, 88 substitution with, 421 Radiocarbon dating, 261 Radius of convergence, 544 Random selection in a circle, 567–568 Index Random variables continuous, 558–565 discrete, 552–558 exponential, 571 with exponential density function, 573 geometric, 582–583 normal, 571, 573 Poisson, 580–582 uniform, 567 Rates of change area under graph of, 323 average, 112–113, 127 derivatives as, 106–107, 112–122 instantaneous, 113, 127 integral of, 304 measurement of, 113 net, 496 partial derivatives and, 356–359 percentage, 288 population model, 113–114 proportional, 461 proportional to quantity, 492 relative, 287 slope as, 62 slope of graph as, 68 time, 208–210 units, 119 Rational equations, solving, 30–31 Rational functions adding, 22 in cost-benefit model, 17–18 defined, 17, 52 derivative of, 87–88 domain, 17 integral of, 422–423 limit of, 85 multiplying, 23 quotients of, 23 Rational numbers, 529 Rationalizing tricks, 85–86 Real numbers, Rectangles approximating regions with, 311 area, 168, 560 illustrated, 41 infinite series representation by, 535 maximizing, 165–166 perimeter, 168 in Riemann sum approximation, 312, 313 Rectangular boxes, 41 Reflections, 240 Regression line, 377 Related rates defined, 217 point moving along a graph, 217 suggestions for problem solutions, 218 weekly sales to price, 218 Relative extreme points defined, 132 locating, 150–152 Relative frequency histogram, 553 table, 553, 554 Relative maximum point, 132 Relative minimum point, 132 Relative rates of change See also Rates of change constant, 273 defined, 271, 287 value of investments example, 272–273 Remainder formula, 518 Retirement account example, 477–478 Revenue business problems, 42–43 change in price versus, 276 curve, 181 demand equation, 182–183 marginal, 109 maximizing, 182, 183 from sales, 44–45 time rate of change of, 209–210 Revenue functions business and economic derivative applications, 181–183 graph of, 44 total, 183 Riemann sum approximation improvement, 313–314 approximation of area of waterfront lot, 312–313 calculation, 311–312 defined, 311 rectangles, 313 Right triangles, 396–397 S Sales, 107–108, 261–262 Scientific notation, 39 Secant function, 409 Secant lines, 78–79, 406 Second derivatives, 105, 126, 142–143 Second order differential equations, 459 Second-derivative rule, 142, 189–190 Second-derivative test applying, 152–154, 159 defined, 152, 190 functions of several variables, 365–366 use selection, 156 Separation of variables defined, 466 initial value problem, 468, 469–470 rewriting left side, 466, 467 solution curves, 468 solutions, 468–469 Sigma notation, 531–532 Signed area, 318 Simplification after quotient rule, 199–200 expressions with logarithms, 248 product rule, 198–199 Simpson’s rule See also Approximation of definite integrals accuracy, 435 defined, 435, 455 geometric interpretation, 436–437 Sine See also Cosine alternative definition of, 397 calculation of, 396 curve, area under, 404 defined, 396 definition diagram, 396 derivatives with, 402 differentiation of, 401–405, 414 general rules, 402 graph of, 398–399 informal justification of differentiation rules for, 405–409 integrals of, 403 integration of, 401–405, 414 in maximizing a volume, 403 in predators-and-prey model, 404–405 properties of, 397–398 secant line approximation for, 406 slope of, 406 in solving right triangle, 397 values of, 396, 397 Slope in applied problems, 56 changing, 134–135 decreasing, 134–135 defined, 57 finding with implicit differentiation, 213–214 increasing, 134–135 as rate of change, 62 of solution curves, 486 of tangent lines, 513 verification of properties, 62–63 Slope fields See also Solutions of differential equations defined, 462 differential equations, 461–463 in solution analysis, 462–463 Slope formulas, 68–69 Slope of curves defined, 67 formulas, 68–70 learning example, 279 at points, 66–73 as rate of change, 68 summary, 124 Slope of graph, 67–70 Slope of lines calculations, 60 commuting, 59 conceptualizing, 57–58 equations of, 57 finding, 60 interpreting, 58–59 linear functions and, 56 negative, 58 parallel, 60 perpendicular, 60 physical interpretation, 58 properties of, 59–60 straight, 57–66 through two points, 61 Slope–intercept equation, 57 Solid boundary, 383 Solids of revolution, 335–336 Solution curves concavity of, 489 slope of, 486 Solutions of differential equations constant, 460 defined, 458 first-order, 474, 475, 476 as functions, 459 initial conditions, 459 initial-value problems, 459–460, 469–470 modeling, 460–461 separation of variables, 468–469 slope fields, 461–463 verifying, 458–459 I-7 I-8 Index Solving differential equations, 459 Standard deviation, 553, 555 Standard normal curve, 449, 574 Standard normal random variables, 574–575 Standard position, angles, 394, 413 Substitution definite integrals with, 429–430 with exponential, 420–421, 422 with fractional powers, 422 integration by, 419–424 with logarithm, 421–422 marginal rate of, 216–217 with radical, 421 with trigonometric functions, 423 Sum rule, 98–102 Sums algebraic operation, 52 approximating with definite integral, 331 of cubes, finding, 31 finite, 532 of functions, differentiating, 99–100 infinite series association, 527 of lengths, 40 partial, 527 Surface area, 41–42 T Table of values, 83 Tangent function defined, 409 derivative of, 410–411 in determining height of buildings, 410 equation, 410 graph of, 411 interpretation of, 409 Tangent lines concavity relationship, 135 to curves, 67 equation of, 76–77 to exponential graph, 232 point–slope equation of, 76 at relative minimum, 361 slope of, 513 t-axis, 501 Taylor polynomials accuracy of approximation, 517–518 to approximate integrals, 516 approximation of ex , 514–515 approximations, 514–516 centering, 517 defined, 514, 546 first, 520 nth, 517 remainder formula and, 518 second, 516 summary, 546—547 at x = a, 517 Taylor series centered at 0, 540–541 defined, 540 definite integrals, 543 differentiation and integration, 541–542 evaluating, 542 expansion, 540, 548 new, from known ones, 542–543 operations on, 541–543 as polynomial of infinite degree, 541 as power series, 540 at x = a, 544 Technology Exercises antidifferentiation, 300 applications of definite integral, 338 applications of first-order differential equations, 484 approximation of definite integrals, 441 areas in xy-plane, 330 compound interest, 271 curve sketching, 163 definite integral and area under a graph, 317 derivative and limits, 82 derivative as rate of change, 122 derivative of ln x, 247 describing graphs of functions, 140 differentiation of exponential functions, 239–240 exponential function ex , 235 exponential functions, 229 exponential models, 286 exponential/normal random variables, 578 exponents and power functions, 40 first-/second-derivative rules, 149 first-/second-derivative tests, 158 functions and graphs, 13 functions and graphs in applications, 49 graphing solutions of differential equations, 491 infinite series, 534 limits and the derivative, 92 more about derivatives, 112 natural logarithm function, 244 natural logarithm function in economics, 277 Newton–Raphson algorithm, 526 numerical solutions of differential equations, 505 optimization problems, 171, 179 Poisson and geometric random variables, 585 product and quotient rules, 205 sine and cosine, 400–401 slope of a curve at a point, 72 slope of a straight line, 66 solutions of differential equations, 465 Taylor polynomials, 519 Temperature, Terminal side, angle, 394, 413 Theory of the firm, 180 Time between phone calls example, 573 Time constant, 262 Time rate of change, 208–210 Trapezoidal rule See also Approximation of definite integrals defined, 433, 454 error from, 435 geometric interpretation, 436–437 Trapezoids, approximation by, 437 Trigonometric functions cosecant, 409 cosine, 395–401 cotangent, 409 differentiation of sine and cosine, 401–405 graphing, 399 informal justification of differentiation rules for sine and cosine, 407–409 product of x and, 426 radian measure of angles, 392–395 secant, 409 sine, 395–401 substitution with, 423 tangent, 409–412 ty-graphs, 488, 489, 493 U Uniform random variables, 567 Units, rate of change, 119 V Variable costs, 15 Variables dependent, different, 104 differentiation with respect to, 104 functions of, 347–391 random, continuous, 558–565 random, discrete, 552–558 separation of, 466–473, 506–507 three, Lagrange multipliers in, 372–373 Variance, 553, 555, 566 Velocity average, 115 constant, 280 rate of change and, 114 Vertical asymptotes, 137 Vertical line test, 9–10 Volume of cones, 336 cube formula, 168 cylinder formula, 168 defined, 41 with double integral, 385 maximizing, 167–168, 403 solid of revolution, 335, 336 of solids, 336 W Weighted averages, 435 Windpipe problem, 175–176 World oil consumption example, 325–326 X x -axis area bounded by a curve and, 319–320 area under, 318 regions above/below, 319 tangent line crossing, 521 x -intercepts computing, 159 defined, 15, 136 determining, 15–16 xy-plane, areas in, 318–331 Y y-intercept defined, 15, 57, 136 finding, 60 interpreting, 58–59 yz -graphs, 488 Z Zeros of functions computing, 32 defined, 26, 53, 136 finding, 27 Useful Algebraic Facts Laws of Exponents Quadratic Formula The solutions of the quadratic equation a ·a = a ax = ax−y ay (ax )y = axy x y x+ y ax2 + bx + c = 0, Laws of Logarithms a=0 are given by √ −b ± b2 − 4ac x= 2a ax · bx = (ab)x ln = ln e = ln xa = a ln x ln xy = ln x + ln y y y = ln x x (1, 0) Factorization and Product Formulas FOIL: (a + b)(c + d) = ac + ad + bc + bd First Outside Inside Last x2 − y = (x + y)(x − y) x3 ± y = (x ± y)(x2 ∓ xy + y ) (x + y)2 = x2 + 2xy + y (x + y)3 = x3 + 3x2 y + 3xy + y Rational Expressions c ad ± bc a ± = b d bd ac a c · = b d bd a d ad a/b = · = c/d b c bc The Natural Exponential Function e = 2.718281 Nature exponential function = ex y y = ex (0, 1) x Useful Geometric Formulas Rectangular Solid Rectangle Right Circular Cone h h h l r w w Perimeter = 2w + 2h Area = wh Volume = lwh Surface area = 2wh + 2wl + 2lh Volume = 13 πr2 h Circle Sphere Trapezoid d r Circumference = 2πr = πd Area = πr2 = 14 πd2 Triangle h2 h1 r w (h1 + h2 ) w Volume = 43 πr3 Surface area = 4πr2 Area = Right Cylinder Pythagorean Theorem c r h h a a2 + b2 = c2 b Area = bh Volume = πr h Surface area = 2πrh (no top or bottom) b Useful Differentiation Facts Secant-Line Approximation of Derivative Definition of Derivative y f (a) = lim h→0 y = f(x) f (a + h) − f (a) h Secant line (a + h, f (a + h)) Tangent line (a, f (a)) x f (a) = slope of tangent line at (a, f (a)) As h → 0, slope of secant line approaches slope of tangent line The Differential One Variable: f (a + h) − f (a) ≈ hf (a) (h close to 0) Several Variables: f (a + h, b + k) − f (a, b) ≈ h ∂f ∂x +k (a,b) ∂f ∂y (a,b) Rules for Differentiation d d d [f (x) ± g(x)] = [f (x)] + [g(x)] Sum Rule: dx dx dx d d [kf (x)] = k · [f (x)] Constant Multiple Rule: dx dx d [f (x)g(x)] = f (x)g (x) + g(x)f (x) Product Rule: dx g(x)f (x) − f (x)g (x) d f (x) = Quotient Rule: dx g(x) [g(x)]2 d f (g(x)) = f (g(x))g (x) Chain Rule: dx dy du dy = , y = f (u) with u = g(x) dx du dx Derivatives of Common Functions d d n (x ) = nxn −1 (sin x) = cos x dx dx d kx d (e ) = kek x (cos x) = − sin x dx dx d d (ln x) = (tan x) = sec2 x dx x dx Exponential Differential Equation If y = f (x) satisfies y = ky, then y = Cek x for some constant C (h, k close to 0) Useful Integration Facts Indefinite Integral f (x) dx = F (x) + C provided that F (x) = f (x) Fundamental Theorem of Calculus Suppose that f (x) is continuous on the interval [a, b] with antiderivative F (x); then b f (x) dx = F (b) − F (a) Riemann Sum Approximation b f (x) dx = lim [f (x1 ) + f (x2 ) + · · · + f (xn )]Δx, Δ x→0 a where xi is from the ith subinterval of [a, b], of length Δx a Midpoint Rule b f (x) dx ≈ [f (x1 ) + · · · + f (xn )]Δx, Integration by Substitution To determine a where xi is the midpoint of the ith subinterval f (g(x))g (x) dx: Integration by Parts f (x)g(x) dx = f (x)G(x) − f (x)G(x) dx, where G(x) is an antiderivative of g(x) Set u = g(x), du = g (x) dx Determine f (u) du = F (u) Substitute the value of u: f (g(x))g (x) dx = F (g(x)) + C Integration Facts xn dx = xn + + C, n = −1 n+1 dx = ln |x| + C x ek x dx = ek x + C k sin x dx = − cos x + C cos x dx = sin x + C sec2 x dx = tan x + C Simpson’s Rule Trapezoidal Rule b b f (x) dx ≈ [f (a0 ) + 4f (x1 ) a + 2f (a1 ) + 4f (x2 ) + 2f (a2 ) + · · · Δx , + 2f (an −1 ) + 4f (xn ) + f (an )] where the are the endpoints and the xi are the midpoints of the subintervals f (x) dx ≈ [f (a0 ) + 2f (a1 ) + · · · a Δx , where the are the endpoints of the subintervals + 2f (an −1 ) + f (an )] ... applied calculus courses T The Series This text is part of a highly successful series consisting of three texts: Finite Mathematics and Its Applications, Calculus and Its Applications, and Brief Calculus. .. thirteenth edition of Calculus and Its Applications, and its Brief version, is written for either a one- or two-semester applied calculus course Although this edition reflects many revisions as requested... applications. ? ?13th ed / Larry J Goldstein [et al.] p cm Includes bibliographical references and index ISBN 0-321-84890-X Calculus? ??Textbooks I Goldstein, Larry Joel II Title: Calculus and its applications