Larson r , hodgkins a college algebra and calculus an applied approach 1ed 2010

ZeroNegative integers Positive integers Section 0.1 Real Numbers: Order and Absolute Value... Points to the left of zero are associated with negative numbers, and points to the right of

Biology and Life Sciences

Blue oak, height of, 103

Body mass index (BMI), 414

E coli bacterium, length, 65

Ecology, fencing a study plot, 743

Endangered species population, 393, 800,

889

Environment

contour map of the ozone hole, 950

oxygen level in a pond, 606, 686

Fruit tree maximum yield, 696, 740

Galloping speeds of animals, 370

Gardening, 825

Genders of children, 1181, 1194, 1229

Genetically modified soybeans, 438

Genetics, 1229

Gestation period of rabbits, 557

Growth of a red oak tree, 686

Gypsy moths, 525

Hardy-Weinberg Law, 976, 985Health

AIDS cases, 1194blood oxygen level, 108, 113blood pressure, 1050body temperature, 137, 595epidemic, 854, 889exposure to a carcinogen and mortality, 1021

exposure to sun, 742infant mortality, 995U.S HIV/AIDS epidemic, 640and wellness, 481

velocity of air flow into and out ofthe lungs, 1050, 1070

Heart rate, 450Human height, 78, 137, 188Hydroflourocarbon emissions, 103Kidney donation, 1164

Lab practical, 1169Litter of kittens, 1230Liver transplants, 268Lung volume, 217Medical sciencedrug concentration, 803length of pregnancy, 1206surface area of a human body, 1019velocity of air during coughing, 668volume of air in the lungs, 867Medicine

amount of drug in bloodstream, 594, 622bone graft procedures, 379

days until recovery after a medicalprocedure, 1206, 1231drug absorption, 910drug concentration in bloodstream,

300, 333, 583, 686, 736, 910, 1150duration of an infection, 976effectiveness of a pain-killing drug, 594healing rate of a wound, 353heart transplants, 1232multiple births, twins, 685Poiseuille’s Law, 686spread of a virus, 388, 678, 801temperature of a patient, 1041treatment of a bacterial infection, 1019Metabolic rate, 113

Nutrition, 447, 986Optimal area of an archaeological digsite, 336

Orthopedic implant sales, 1107Oxygen level, 61

Peregrine falcons, 450Pest management in a forest, 191Physiology

blood flow, 845body surface area, 736

Plant biology lab, 1169Plant growth, 1060Population

794, 890, 900Predator-prey cycle, 1046, 1050, 1051Psychology

Ebbinghaus Model, 767human memory model, 332, 333, 339,

360, 362, 381, 399, 400, 402, 880intelligence quotient (IQ), 1233

IQ scores, 136, 403learning curve, 392, 401, 718, 795learning theory, 759, 767, 777, 781,

786, 1195, 1205memory experiment, 898, 900, 927migraine prevalence, 580

skill retention model, 363sleep patterns, 868Stanford-Binet Test, (IQ test), 967Ratio of reptiles, 1089

Research study, 19Respiratory diseases, 1174Stocking a lake with fish, 392, 976Suburban wildlife, 381

Systolic blood pressure, 604Tree growth, 816

Water pollution, 332Weight of a puppy, 182, 189Weights of adult male rhesus monkeys,1203

Wheelchair ramp, 181Wildlife management, 401, 718, 736Zebrafish embryos, 1229

Business and EconomicsAdvertising, 192expenses, 278, 297, 301, 303, 648,

986, 1150Annualoperating cost, 136payroll of new car dealerships, 1089sales, 78, 251, 338, 352, 414, 450,1107

Average cost, 205, 206, 332, 339, 677,

705, 714, 716, 718, 840, 1150Average cost and profit, 742

Average production, 1012

Average profit, 718, 1010

Average revenue, 1012

Average weekly demand, 1197

Average weekly profit, 1012

Book value per share, 205, 217

Cash flow per share, Harley-Davidson, 791

Charter bus fares, 206

Cobb-Douglas production function, 640,

Dow Jones Industrial Average, 596, 678

Earnings-dividend ratio, Wal-Mart

Stores, 247

Earnings per share, 170, 929, 956

equity, PepsiCo, 996Economics, 595

equation of exchange, 1018gross domestic product, 736investment, 1195

marginal benefits and costs, 816present value, 926

revenue, 740Elasticity of demand, 706, 719, 741Elasticity and revenue, 703Equimarginal Rule, 985Expected sales, 1185Factory production, 494, 530Federal

cost of food stamps, 207debt, 438

financial aid awarded, 254government expenses, 19Pell Grants, 47, 414Perkins Loans, 414student aid, 47Finance, cyclical stocks, 1051Flour production, 125Fuel cost, 851Furniture production, 449Gold prices, 169, 217, 267Hotel pricing, 494Increasing production, 646Increasing profit, 143Insurance, 1194Inventorycost, 677, 741

of digital cameras, 1230

of kayaks, 449levels, 494, 530, 533

of liquefied petroleum gases, 1069management, 557, 596

of movie players, 464replenishment, 607Job applicants, 1168, 1170, 1229Labor/wage requirements, 495, 507, 530Least-Cost Rule, 985

Lifetime of a product, 1191Making a sale, 1179, 1181, 1182Managing a store, 607

Manufacturing, 1206Marginal analysis, 731, 732, 736, 742,

845, 909Marginal cost, 594, 595, 596, 624, 705,

716, 833, 966, 1019Marginal productivity, 966Marginal profit, 588, 592, 594, 595, 596,

624, 705Marginal revenue, 591, 594, 595, 624,

705, 966, 1019Market analysis, 1195Market research, 122, 125, 152

Marketing, 889Maximum production level, 980, 981,

1019, 1021Maximum profit, 218, 666, 701, 705,

706, 719, 972, 982Maximum revenue, 698, 700, 705, 706, 764Mean and median useful lifetimes of aproduct, 1200

Media selection for advertising, 459Minimum average cost, 699, 705, 719,

785, 786Minimum cost, 695, 696, 697, 706, 740,

977, 1019Mobile homes manufactured, 205Monthly cost, 103, 114

Monthly profit, 87Monthly sales, 92Monthly flight cost, 124National defense budget, 205National defense outlays, 332National deficit, 675Negotiating a price, 606Number of Kohl’s stores, 901Office space, 986

Optimalcost, 267, 336, 456, 459profit, 267, 336, 455, 458, 459, 460,

465, 466revenue, 266, 336, 459, 465Owning a franchise, 557Patents issued, 200Payroll mix-up, 1181Point of diminishing returns, 675, 677Point of equilibrium, 422, 425, 440, 462, 466Price-earning (P/E) ratio, 237

Price of a product, 153Production, 264, 640, 865, 952, 955, 985cost, 102

limit, 91Productivity, 677

of a new employee, 363Profit, 154, 155, 170, 227, 247, 268,

Real estate, 1020Reimbursed expenses, 191Returning phone calls, 1225

of prescription drugs by mail order, 153

Procter & Gamble, 707

Sales, equity, and earnings per share,

Johnson & Johnson, 1018

Shareholder’s equity, Wal-Mart, 956, 967

Social Security Trust Fund, 854

State income tax, 183

State sales tax, 183, 190

Weekly demand, 1190Worker’s productivity, 395Years of service for employees, 1176Interest Rates

Annuity, 842, 845, 867Balance in an account, 25, 28, 39, 65,

67, 252, 372, 378, 390, 402, 754,

756, 1159Becoming a millionaire, 28Bond investment, 505Borrowing money, 124, 436, 481, 529Cash advance, 124, 152

Cash settlement, 351Certificate of deposit, 759Charitable foundation, 921College tuition fund, 880Comparing investment returns, 90Compound interest, 26, 47, 91, 121,

Endowment, 921Finance, 777present value, 926Future value, 758, 880Inflation rate, 26, 750, 768, 803Investment, 382, 449, 463, 466, 956, 967mix, 90, 414

plan, 398portfolio, 424, 425, 433, 437, 460, 463Rule of 70, 794

strategy, 986time, 362, 399Monthly payments, 61, 953, 956Present value, 352, 756, 758, 801, 876,

877, 880, 901, 909, 921, 926, 928

of a perpetual annuity, 919Savings plan, 44, 47

Scholarship fund, 921Simple interest, 84, 89, 91, 135, 150, 181,

190, 407, 466Stock mix, 90Tripling time, 776Trust fund, 758

Acceleration, 629, 649Acceleration due to gravity, 630Accuracy of a measurement, 133, 137, 152Acid mixture, 462

Acid solution, 87Acidity of rainwater, 1018Airplane speed, 421, 424Automobile aerodynamics, 227Automobile crumple zones, 382Average velocity, 584

Biomechanics, Froude number, 1018Boiling temperature of water, 785Bouncing ball, 1116

Capacitance in series circuits, 92Carbon dating, 386, 392, 777, 794Catenary, 763

Charge of an electron, 28Chemical reaction, 395Chemistry experiment, 1229Circuit analysis, 505Comet orbit, A25, A28Dating organic material, 746Diesel mechanics, 247Earth and its shape, 947, 977Earthquake magnitude, Richter scale,

389, 395, 401, 786Electricity, 684Electron microscopes, 28Escape velocity, 35, 38Estimating speed, 38Estimating the time of death, 395Falling object, 97, 101, 112, 114, 1098Eiffel tower, 111

Grand Canyon, 151instantaneous rate of change, 585

on the moon, 151the owl and the mouse, 101Royal Gorge Bridge, 101Fluid flow, A28

Geologycontour map of seismic amplitudes, 956crystals, 938

Hot air balloon, 112Hydrogen orbitals, 1232Ideal Gas Law, 91Kinetic energy, 91Lensmaker’s equation, 92Measurement errors, 734, 736Metallurgy, 1205

Meteorologyamount of rainfall, 1195annual snowfall in Reno, Nevada, 193atmospheric pressure, 956

average monthly precipitationfor Bismarck, North Dakota, 1070for Sacramento, California, 1069for San Francisco, California, 1077

(continued on back endsheets)

College Algebra and Calculus

An Applied Approach

R O N L A R S O N

The Pennsylvania State University

The Behrend College

A N N E V H O D G K I N S

Phoenix College

with the assistance of

D AV I D C F A LV O

The Pennsylvania State University

The Behrend College

Ron Larson and Anne V Hodgkins

VP/Editor-in-Chief: Michelle Julet

Publisher: Richard Stratton

Senior Sponsoring Editor: Cathy Cantin

Associate Editor: Jeannine Lawless

Editorial Assistant: Amy Haines

Associate Media Editor: Lynh Pham

Senior Content Manager: Maren Kunert

Marketing Specialist: Ashley Pickering

Marketing Assistant: Angela Kim

Marketing Communications Manager:

Mary Anne Payumo

Associate Content Project Manager, Editorial Production:

Jill Clark

Art and Design Manager: Jill Haber

Senior Manufacturing Buyer: Diane Gibbons

Senior Rights Acquisition Account Manager: Katie Huha

Text Designer: Jean Hammond

Art Editor: Larson Texts, Inc.

Senior Photo Editor: Jennifer Meyer Dare

Illustrator: Larson Texts, Inc.

Cover Designer: Sarah Bishins

Cover Image: © Vlad Turchenko, istockphoto

Compositor: Larson Texts, Inc.

ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used

in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher.

Library of Congress Control Number: 2008929478 ISBN-13: 978-0-547-16705-3

ISBN-10: 0-547-16705-9

Brooks/Cole

10 Davis Drive Belmont, CA 94002 USA

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1 2 3 4 5 6 7 12 11 10 09 08

A Word from the Authors (Preface) viiiTextbook Features x

Polynomial and Rational Functions 256

Limits and Derivatives 534

Integration and Its Applications 805

Mid-Chapter Quiz

14.5 Integrals of Trigonometric FunctionsChapter Summary and Study StrategiesReview Exercises

Series and Taylor Polynomials (online)*

Mid-Chapter Quiz

Chapter Summary and Study Strategies

Review Exercises

Chapter Test

Appendix A: An Introduction to Graphing Utilities

Appendix B: Conic Sections

Appendix C: Further Concepts in Statistics

Appendix D: Alternative Introduction to the Fundamental

Theorem of Calculus

Appendix E: Formulas

Appendix F: Differential Equations

Appendix G: Properties and Measurement

Appendix H: Graphing Utility Programs

Welcome to the first edition of College Algebra and Calculus: An Applied Approach! This textbook completes the publication of a whole series of textbooks

tailored to the needs of college algebra and applied calculus students majoring in business, life science, and social science courses

Many students take college algebra as a prerequisite for applied calculus We wrote all of these books using the same design, writing style, and pedagogical features, with the goal of providing these students with a level of familiarity that encourages confidence and a smooth transition between the courses Additionally, by combining the college algebra and applied calculus material into one textbook, we have given students one comprehensive resource for both courses.

We’re excited about this new textbook because it acknowledges where students are when they enter the course— and where they should be when they complete

it We review the basic algebra that students have studied previously (in Chapter 0 and in the exercises, notes, study tips and algebra review notes throughout the

text), and present solid college algebra and applied calculus courses that balance

understanding of concepts with the development of strong problem-solving skills

In addition, emphasis was placed on providing an abundance of real-world problems throughout the textbook to motivate students’ interest and understanding Applications were taken from news sources, current events, government data, and industry trends to illustrate concepts and show the relevance of the math.

We hope you and your students enjoy College Algebra and Calculus: An Applied Approach We are excited about this new textbook program because it helps

students learn the math in the ways we have found most effective for our

students — by practicing their problem-solving skills and reinforcing their

understanding in the context of actual problems they may encounter in their lives and careers.

Please do tell us what you think Over the years, we have received many useful ments from both instructors and students, and we value these comments very much.

com-Ron Larson

Anne V Hodgkins

A Word from the Authors

College Algebra with Applications for Business and the Life Sciences

Calculus: An Applied Approach, Eighth Edition Brief Calculus: An Applied Approach, Eighth Edition Applied Calculus for the Life and Social Sciences

College Algebra and Calculus: An Applied Approach

Thank you to the many instructors who reviewed College Algebra with Applications for Business and the Life Sciences, Calculus: An Applied Approach Eighth Edition, and Brief Calculus: An Applied Approach Eighth Edition, and

encouraged us to try something new Without their help, and the many suggestions

we’ve received throughout the previous editions of Calculus: An Applied Approach, this book would not have been possible Our thanks also to Robert

Hostetler, The Behrend College, The Pennsylvania State University, and Bruce Edwards, University of Florida, for their significant contributions to previous editions of this text.

Reviewers of College Algebra with Applications for Business and the Life Sciences

Michael Brook, University of Delaware Tim Chappell, Metropolitan Community College—Penn Valley Warrene Ferry, Jones County Junior College

David Frank, University of Minnesota Michael Frantz, University of La Verne Linda Herndon, OSB, Benedictine College Ruth E Hoffman, Toccoa Falls College Eileen Lee, Framingham State College Shahrokh Parvini, San Diego Mesa College Jim Rutherfoord, Chattahoochee Technical College

Reviewers of the Eighth Edition of Calculus: An Applied Approach

Lateef Adelani, Harris-Stowe State University, Saint Louis; Frederick Adkins, Indiana University of Pennsylvania; Polly Amstutz, University of Nebraska at Kearney; Judy Barclay, Cuesta College; Jean Michelle Benedict, Augusta State University; Ben Brink, Wharton County Junior College; Jimmy Chang, St Petersburg College; Derron Coles, Oregon State University; David French, Tidewater Community College; Randy Gallaher, Lewis & Clark Community College; Perry Gillespie, Fayetteville State University; Walter J Gleason, Bridgewater State College; Larry Hoehn, Austin Peay State University; Raja Khoury, Collin County Community College; Ivan Loy, Front Range Community College; Lewis D Ludwig, Denison University; Augustine Maison, Eastern Kentucky University; John Nardo, Oglethorpe University; Darla Ottman, Elizabethtown Community & Technical College; William Parzynski, Montclair State University; Laurie Poe, Santa Clara University; Adelaida Quesada, Miami Dade College – Kendall; Brooke P Quinlan, Hillsborough Community College; David Ray, University of Tennessee at Martin; Carol Rychly, Augusta State University; Mike Shirazi, Germanna Community College; Rick Simon, University of La Verne; Marvin Stick, University of Massachusetts – Lowell; Devki Talwar, Indiana University of Pennsylvania; Linda Taylor, Northern Virginia Community College; Stephen Tillman, Wilkes University; Jay Wiestling, Palomar College; John Williams, St Petersburg College; Ted Williamson, Montclair State University

Acknowledgments

How to get the most out of your textbook

Establish a Solid Foundation

(See Section 8.1, Exercise 45.)

Derivatives have many real-life applications The applications listed below represent a sample of the applications in this chapter.

Applications

8.1 Higher-Order Derivatives

8.2 Implicit Differentiation

8.3 Related Rates

8.4 Increasing and Decreasing Functions

8.5 Extrema and the First-Derivative Test

8.6 Concavity and the Test

■ Determine if an equation or a set of ordered pairs represents a function.

■ Use function notation and evaluate a function.

■ Find the domain of a function.

■ Write a function that relates quantities in an application problem.

194 C H A P T E R 2 Functions and Graphs

Section 2.4

Functions

Definition of a Function

A function from a set to a set is a rule of correspondence that assigns

to each element in the set exactly one element in the set The set is

the domain (or set of inputs) of the function and the set contains the

range (or set of outputs).

B f,

A B.

y A

x

B A f

Vertical Line Test for Functions

A set of points in a coordinate plane is the graph of as a function of if and only if no vertical line intersects the graph at more than one point.

x y

Example 7 The Path of a Baseball

A baseball is hit 3 feet above home plate at a velocity of 100 feet per second and

an angle of The path of the baseball is given by the function

where and are measured in feet Will the baseball clear a 10-foot fence located

300 feet from home plate?

SOLUTION When the height of the baseball is given by

The ball will clear the fence, as shown in Figure 2.42.

F I G U R E 2 4 2

Notice that in Figure 2.42, the baseball is not at the point before it is hit.

This is because the original problem states that the baseball was hit 3 feet above

A bulleted list of learning

objectives enables you to preview

what will be presented in the

upcoming section.

C H A P T E R O P E N E R S

Each opener has an applied example of

a core topic from the chapter The section

outline provides a comprehensive

overview of the material being presented.

D E F I N I T I O N S A N D T H E O R E M S

All definitions and theorems are highlighted

for emphasis and easy recognition.

E X A M P L E S

There is a wide variety of relevant examples in the text, each titled for easy reference Many of the solutions are presented graphically, analytically, and/or numerically to provide further insight into mathematical concepts Examples based on a real-life situation are identified with an icon

1 Determine whether the following statement is true or false Explain your reasoning.

The points and both lie on the same circle whose center is the origin.

2 Explain how to find the - and -intercepts of the graph of an equation.

3 For every point on a graph, the point is also on the graph What type of symmetry must the graph have? Explain.

4 Is the point on the circle whose equation in standard form is

冇ⴚ4, 3冈

冇3, 4冈

C O N C E P T C H E C K

✓CHECKPOINT 4

Evaluate the function in

Example 4 when x  3 and 3. ■

S T U D Y T I P

When applying the properties

of logarithms to a logarithmic function, you should be careful

to check the domain of the function For example, the

whereas the domain of

These noncomputational questions

appear at the end of each section

and are designed to check your

understanding of the key concepts.

C H E C K P O I N T

After each example, a similar problem is presented to allow for immediate practice and to provide reinforcement of the concepts just learned.

S T U D Y T I P S

Scattered throughout the text, study tips address

special cases, expand on concepts, and help you to

avoid common errors.

S K I L L S R E V I E W

These exercises at the beginning of each exercise set help you review skills covered

in previous sections The answers are provided at the back of the text to reinforce understanding

of the skill sets learned.

S E C T I O N 2 7 The Algebra of Functions 235

In Exercises 29–36, find (a) and (b)

In Exercises 41– 44, use the graphs of and to

evaluate the functions.

41 (a) (b)

42 (a) (b)

43 (a) (b)

44 (a) (b)

In Exercises 45–52, find two functions and such that

(There are many correct answers.)

54 CostThe weekly cost of producing units in a manufacturing process is given by the function The number of units produced in hours is given by Find and interpret

55 CostThe weekly cost of producing units in a manufacturing process is given by the function The number of units produced in hours is given by Find and interpret

56 Comparing ProfitsA company has two manufacturing plants, one in New Jersey and the other in California From New Jersey were decreasing according to the function where represents the profits (in millions of dollars) and represents the year, with corresponding to 2000 On the other hand, the profits for the manufacturing plant in California were increasing according to the function Write a function that represents the overall company

profits during the nine-year period Use the stacked bar

the company during this nine-year period, to determine decreasing.

5.00 10.00 25.00 45.00

0 2 4 6 8 Year (0 ↔ 2000)

P1 P2

t P

C 共x兲  50x  495.

x C

共Cx 兲共t兲.

x 共t兲  40t.

t x

C 共x兲  70x  800.

x C

Mid-Chapter Quiz See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Take this quiz as you would take a quiz in class When you are done, check your work against the answers given in the back of the book.

In Exercises 1 and 2, sketch the graph of the quadratic function Identify the vertex and the intercepts.

1.

In Exercises 3 and 4, describe the right-hand and left-hand behavior of the graph of the polynomial function Verify with a graphing utility.

3.

5 Use synthetic division to evaluate when

In Exercises 6 and 7, write the function in the form for the given value of and demonstrate that

13 The profit (in dollars) for a clothing company is

where is the advertising expense (in tens of thousands of dollars) What is the profit for an advertising expense of $450,000? Use a graphing utility to approximate another advertising expense that would yield the same profit.

14 CropsThe worldwide land areas A (in millions of hectares) of transgenic crops

for the years 1996 to 2006 are shown in the table.(Source: International Service for the Acquisition of Agri-Biotech Applications)

(a) Use a graphing utility to create a scatter plot of the data Let represent the year, with corresponding to 1996.

(b) Use the regression feature of a graphing utility to find a linear model, a quadratic

model, a cubic model, and a quartic model for the data.

(c) Use a graphing utility to graph each model separately with the data in the same viewing window How well does each model fit the data?

(d) Use each model to predict the year in which the land area will be about

150 million hectares Explain any differences in the predictions.

In Exercises 1 and 2, find the distance between the points and the midpoint

of the line segment connecting the points.

3 Find the intercepts of the graph of

4 Describe the symmetry of the graph of

5 Find an equation of the line through with a slope of

6 Write the equation of the circle in standard form and sketch its graph.

In Exercises 7 and 8, decide whether the statement is true or false Explain.

7 The equation identifies as a function of

8 If and the represents a function from to

In Exercises 9 and 10, (a) find the domain and range of the function, (b) determine the intervals over which the function is increasing, decreasing, or imate any relative minimum or relative maximum values of the function.

9. (See figure.) 10. (See figure.)

In Exercises 11 and 12, sketch the graph of the function.

17 A business purchases a piece of equipment for $30,000 After 5 years, the equipment

will be worth only $4000 Write a linear equation that gives the value V of the

equip-ment during the 5 years.

18 PopulationThe projected populations (in millions) of children under the age of

5 in the United States for selected years from 2010 to 2050 are shown in the table Use data Let represent the year, with corresponding to 2010.(Source: U.S Census Bureau)

2x  3y  5

x2 y2 6x  4y  3  0

2 共3, 5兲

y x

2  4

y共x  5兲共x  3兲.

共3.25, 7.05兲, 共2.37, 1.62兲 共3, 2兲, 共5, 2兲

Chapter Test See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Figure for 9

−2

6 2

−4 −2 4

x y

■ Find the domain of a rational function. 89–92

■ Find the vertical and horizontal asymptotes of the graph of a rational function. 89–92

Let

1 The graph of has vertical asymptotes at the zeros of

2 The graph of has one or no horizontal asymptote determined by comparing

the degrees of and

a If the graph of has the line (the -axis) as a horizontal asymptote.

b If the graph of has the line (ratio of the leading coefficients) as a horizontal asymptote.

c If the graph of has no horizontal asymptote.

■ Sketch the graph of a rational function, including graphs with slant asymptotes. 93–98

■ Use a rational function model to solve an application problem. 99–103

Chapter Summary and Study Strategies 335

Section 3.5 Review Exercises

■ Find the complex conjugate of a complex number. 49–52

■ Perform operations with complex numbers and write the results in standard form. 53– 68

■ Solve a polynomial equation that has complex solutions. 69–72

■ Plot a complex number in the complex plane. 73, 74

共a  bi兲共c  di兲  共ac  bd兲  共ad  bc兲i

共a  bi兲  共c  di兲  共a  c兲  共b  d兲i

Section 3.6

■ Use the Fundamental Theorem of Algebra and the Linear Factorization 75–80

Theorem to write a polynomial as the product of linear factors.

■ Find a polynomial with real coefficients whose zeros are given. 81, 82

■ Factor a polynomial over the rational, real, and complex numbers. 83, 84

■ Find all real and complex zeros of a polynomial function. 85–88

Study Strategies

■Use a Graphing UtilityA graphing calculator or graphing software for a computer can help you in this course in

two important ways As an exploratory device, a graphing utility allows you to learn concepts by allowing you to compare

graphs of functions For instance, sketching the graphs of and helps confirm that the negative

coefficient has the effect of reflecting the graph about the -axis As a problem-solving tool, a graphing utility frees you

from some of the difficulty of sketching complicated graphs by hand The time you can save can be spent using mathematics

to solve real-life problems.

■Problem-Solving StrategiesIf you get stuck when trying to solve a real-life problem, consider the strategies below.

1 Draw a Diagram If feasible, draw a diagram that represents the problem Label all known values and unknown values on

the diagram.

2 Solve a Simpler Problem Simplify the problem, or write several simple examples of the problem For instance, if you are

asked to find the dimensions that will produce a maximum area, try calculating the areas of several examples.

3 Rewrite the Problem in Your Own Words Rewriting a problem can help you understand it better.

4 Guess and Check Try guessing the answer, then check your guess in the statement of the original problem By refining

your guesses, you may be able to think of a general strategy for solving the problem.

M I D - C H A P T E R Q U I Z

Appearing in the middle of each chapter, this one-page test allows you to practice skills and concepts learned in the chapter This opportunity for self-assessment will uncover any potentially weak areas that might require further review of the material.

C H A P T E R T E S T

Appearing at the end of each chapter, this test is designed to

simulate an in-class exam Taking these tests will help you to

determine what concepts require further study and review.

C H A P T E R S U M M A RY A N D

S T U D Y S T R AT E G I E S

The Summary reviews the skills covered in the

chapter and correlates each skill to the Review

Exercises that test the skill Following each

Chapter Summary is a short list of Study

Strategies for addressing topics or situations

specific to the chapter.

Enhance Your Understanding Using Technology

There are several ways to use your graphing utility to locate the zeros

of a polynomial function after listing the possible rational zeros You

can use the table feature by setting the increments of x to the smallest

difference between possible rational zeros, or use the table feature in ASK

mode In either case the value in the function column will be 0 when x is a

zero of the function Another way to locate zeros is to graph the function

Be sure that your viewing window contains all the possible rational zeros

T E C H N O L O G Y

g p

92 Revenue A company determines that the total revenue (in hundreds of thousands of dollars) for the years 1997

to 2010 can be approximated by the function

where represents the year, with corresponding to

1997 Graph the revenue function using a graphing utility

and use the trace feature to estimate the years during which

the revenue was increasing and the years during which therevenue was decreasing

t 7

t

7 ≤t ≤20

R  0.025t3 0.8t2 2.5t  8.75, R

B u s i n e s s C a p s u l e

SunPower Corporation develops and Power’s new higher efficiency solar cells generate up to 50% more power than other developed by Dr Richard Swanson and his students while he was Professor of Engineering nues are projected to increase 300% from its

manu-2005 revenues.

Internet, or some other reference source to find experiencing strong growth similar to the example small business.

AP/Wide World Photos

117. MAKE A DECISION You are a sales representative for anautomobile manufacturer You are paid an annual salaryplus a bonus of 3% of your sales over $500,000 Considerthe two functions given by

g 共x兲  0.03x.

f 共x兲  x  500,000

共 兲共 兲

M A K E A D E C I S I O N

These multi-step exercises reinforce your problem-solving

skills and mastery of concepts, and take a real-life application

further by testing what you know about a given problem to

make a decision within the context of the problem.

T E C H N O L O G Y E X E R C I S E S

Technology can help you visualize the math and develop a

deeper understanding of mathematical concepts Many of

the exercises in the text can be solved using technology—

giving you the opportunity to practice using these tools

The symbol identifies exercises for which you are

specifically instructed to use a graphing calculator or a

computer algebra system to solve the problem Additionally,

the symbol denotes exercises best solved by using a

Prepare for Success in

Applied Calculus and Beyond

values.

60 Solar Energy Photovoltaic cells convert light energy into electricity The photovoltaic cell and module domestic shipments (in peak kilowatts) for the years 1996 to 2005 are shown in the table. (Source: Energy Information Administration)

(a) Use a spreadsheet software program to create a scatter plot of the data Let represent the year, with corresponding to 1996.

(b) Use the regression feature of a spreadsheet software

program to find a cubic model and a quartic model for the data.

(c) Use each model to predict the year in which the shipments will be about 1,000,000 peak kilowatts.

Then discuss the appropriateness of each model for predicting future values.

0

Fundamental Concepts

of Algebra

The Iditarod Sled Dog Race includes a stop in McGrath, Alaska Part of the

challenge of this event is facing temperatures that reach well below zero

To find the range of a set of temperatures, you must find the distance between

two numbers (See Section 0.1, Exercise 81.)

The fundamental concepts of algebra have many real-life

applications The applications listed below represent a sample

of the applications in this chapter.

■ College Costs, Exercise 75, page 28

Applications

and Absolute Value

■ Classify real numbers as natural numbers, integers, rational numbers, orirrational numbers.

■ Order real numbers

■ Give a verbal description of numbers represented by an inequality

■ Use inequality notation to describe a set of real numbers

■ Interpret absolute value notation

■ Find the distance between two numbers on the real number line

■ Use absolute value to solve an application problem

Real Numbers

The formal term that is used in mathematics to refer to a collection of objects is

the word set For instance, the set

contains the three numbers 1, 2, and 3 Note that a pair of braces is used to enclose the members of the set In this text, a pair of braces will always indicate the members of a set Parentheses ( ) and brackets [ ] are used to represent other ideas.

The set of numbers that is used in arithmetic is the set of real numbers The

term real distinguishes real numbers from imaginary or complex numbers.

A set A is called a subset of a set B if every member of A is also a member

of B Here are two examples.

One of the most commonly used subsets of real numbers is the set of

natural numbers or positive integers

Set of positive integers

Note that the three dots indicate that the pattern continues For instance, the set also contains the numbers 5, 6, 7, and so on.

Positive integers can be used to describe many quantities that you encounter

in everyday life—for instance, you might be taking four classes this term, or you might be paying $700 a month for rent But even in everyday life, positive inte- gers cannot describe some concepts accurately For instance, you could have a zero balance in your checking account, or the temperature could be (10 degrees below zero) To describe such quantities, you need to expand the set of

positive integers to include zero and the negative integers The expanded set is called the set of integers, which can be written as follows.

ZeroNegative integers Positive integers

Section 0.1

Real Numbers:

Order and

Absolute Value

The set of integers is a subset of the set of real numbers This means that every integer is a real number.

Even with the set of integers, there are still many quantities in everyday life that you cannot describe accurately The costs of many items are not in whole dollar amounts, but in parts of dollars, such as $1.19 or $39.98 You might work

hours, or you might miss the first half of a movie To describe such quantities,

the set of integers is expanded to include fractions The expanded set is called the set of rational numbers Formally, a real number is called rational if it can be

written as the ratio of two integers, where (The symbol means not

equal to.) For instance,

and are rational numbers Real numbers that cannot be written as the ratio of two

integers are called irrational For instance, the numbers

and are irrational numbers The decimal representation of a rational number is either

terminating or repeating For instance, the decimal representation of

rounded to a certain number of decimal places For instance, rounded to four

decimal places, the decimal approximations of and are

and The symbol means approximately equal to.

The Venn diagram in Figure 0.1 shows the relationships between the real numbers and several commonly used subsets of the real numbers.

F I G U R E 0 1

IntegersRational Numbers Irrational Numbers

Whole NumbersNatural Numbers

39 100

−

3 7

3 5

Make sure you understand that

not all fractions are rational

numbers For instance, the

fraction is not a rational

number.

冪2

3

The Real Number Line and Ordering The picture that is used to represent the real numbers is the real number line It consists of a horizontal line with a point (the origin) labeled as 0 (zero) Points

to the left of zero are associated with negative numbers, and points to the right

of zero are associated with positive numbers, as shown in Figure 0.2 The real

number zero is neither positive nor negative So, when you want to talk about

real numbers that might be positive or zero, you can use the term nonnegative real

numbers.

F I G U R E 0 2 The Real Number Line

Each point on the real number line corresponds to exactly one real number, and each real number corresponds to exactly one point on the real number line,

as shown in Figure 0.3 The number associated with a point on the real number

line is the coordinate of the point.

F I G U R E 0 3 Every real number corresponds to a point on the real number line.The real number line provides you with a way of comparing any two real numbers For instance, if you choose any two (different) numbers on the real number line, one of the numbers must be to the left of the other number The

number to the left is less than the number to the right, and the number to the right

is greater than the number to the left.

F I G U R E 0 4 a is to the left of b.

a < b

3 2 1

1 0

570

2 3

5

Definition of Order on the Real Number Line

If the real number lies to the left of the real number on the real number line, is less than which is denoted by

as shown in Figure 0.4 This relationship can also be described by saying that

less than or equal to and the inequality means that is greater than

or equal to a.

b

b ≥ a b,

a

a ≤ b

b > a.

a b

a < b

b, a

b a

Origin

Negative numbers Positive numbers

The symbols and are called inequality symbols Inequalities are

useful in denoting subsets of real numbers, as shown in Examples 1 and 2.

Example 1 Interpreting Inequalities

a The inequality denotes all real numbers that are less than or equal to 2,

as shown in Figure 0.5(a).

b The inequality means that and This double

inequality denotes all real numbers between and 3, including but not

including 3, as shown in Figure 0.5(b).

c The inequality denotes all real numbers that are greater than as shown in Figure 0.5(c).

Give a verbal description of the subset of real numbers represented by ■

In Figure 0.5, notice that a bracket is used to include the endpoint of an interval and a parenthesis is used to exclude the endpoint.

Example 2 Inequalities and Sets of Real Numbers

a “ is nonnegative” means that is greater than or equal to zero, which you can

write as

b “ is at most 5” can be written as

c “ is negative” can be written as and “ is greater than ” can be written as Combining these two inequalities produces

d “ is positive” can be written as and “ is not more than 6” can be written as Combining these two inequalities produces

The following property of real numbers is called the Law of Trichotomy As the

“tri” in its name suggests, this law tells you that for any two real numbers and

precisely one of three relationships is possible.

Absolute Value and Distance

The absolute value of a real number is its magnitude, or its value disregarding its

sign For instance, the absolute value of 3, written ⱍ 3 ⱍ , has the value of 3.

a > b

a  b,

a < b, b,

Definition of Absolute Value

Let be a real number The absolute value of , denoted by is

ⱍ a ⱍ  冦 a, if a ≥ 0

a, if a < 0 .

ⱍ a ⱍ ,

a a

Use inequality notation to describe

each subset of real numbers.

Be sure you see from the

defini-tion that the absolute value of a

real number is never negative.

ⱍ 5 ⱍ   共5兲  5 a  5,

The absolute value of any real number is either positive or zero Moreover, 0

is the only real number whose absolute value is zero That is,

Example 3 Finding Absolute Value

SOLUTION

Absolute value can be used to define the distance between two numbers on the real number line To see how this is done, consider the numbers and 4, as shown in Figure 0.6 To find the distance between these two numbers, subtract either number from the other and then take the absolute value of the difference (Distance between and 4)

F I G U R E 0 6 The distance between 3and 4 is 7

4 3 2 1 0

3

43

ⱍ 7 ⱍ  7

ⱍ 0 ⱍ  0.

Properties of Absolute Value

Let and be real numbers Then the following properties are true.

Place the correct symbol

between the two

Distance Between Two Numbers

Let and be real numbers The distance between a and b is given by

Distance  ⱍ b  a ⱍ  ⱍ a  b ⱍ

b a

Example 5 Finding the Distance Between Two Numbers

a The distance between 2 and 7 is

b The distance between 0 and is

c The statement “the distance between and 2 is at least 3” can be written as

Application

Example 6

MAKE A DECISION Budget Variance

You monitor monthly expenses for a home health care company For each type of expense, the company wants the absolute value of the difference between the

actual and budgeted amounts to be less than or equal to $500 and less than or

equal to 5% of the budgeted amount By letting represent the actual expenses and the budgeted expenses, these restrictions can be written as

and For travel, office supplies, and wages, the company budgeted $12,500, $750, and

$84,600 The actual amounts paid for these expenses were $12,872.56, $704.15, and $85,143.95 Are these amounts within budget restrictions?

restrictions is to create the table shown.

Budgeted Actual Expense, b Expense, a 0.05b

In Example 6, the company

budgeted $28,000 for medical

supplies, but actually paid

$30,100 Is this within budget

restrictions? ■

Is the statement true? If not, explain why

1 There are no integers in the set of rational numbers

2 The set of integers is a subset of the set of natural numbers

3 The expression describes a subset of the set of rational numbers

4 When is a negative, ⱍaⱍⴝ ⴚa.

x < 5

C O N C E P T C H E C K

Math plays an important part

in keeping your personal finances

in order as well as a company’s

expenses and budget

SuperStock/Jupiter Images

The symbol indicates an example that uses or is derived from real-life data

In Exercises 1–6, determine which numbers in the

set are (a) natural numbers, (b) integers, (c) rational

numbers, and (d) irrational numbers

In Exercises 7–10, use a calculator to find the

decimal form of the rational number If the number

is a nonterminating decimal, write the repeating

pattern

In Exercises 11 and 12, approximate the two plotted

numbers and place the correct symbol

between them

11.

12.

In Exercises 13–18, plot the two real numbers on the

real number line and place the appropriate inequality

In Exercises 19–22, use a calculator to order the

numbers from least to greatest

39 y is greater than 5 and less than or equal to 12.

40 m is at least and at most 9

41 The person’s age A is at least 35.

42 The yield Y is no more than 42 bushels per acre.

43 The annual rate of inflation r is expected to be at least

3.5%, but no more than 6%

44 The price p of unleaded gasoline is not expected to go

below $2.13 per gallon during the coming year

In Exercises 45–54, evaluate the expression

220,1.7320,

冪3,

26

15,

12790

14

111

940

In Exercises 55–60, place the correct symbol

between the two real numbers

In Exercises 71–78, use absolute value notation to

describe the sentence

71 The distance between and is greater than 1.

72 The distance between and 5 is no more than 3.

73 The distance between and is at least 6

74 The distance between and 0 is less than 8.

75. is at least six units from 0

76. is less than eight units from 0

77. is more than five units from

78. is at most two units from

you pass milepost 57 near Pittsburgh, then milepost 236

near Gettysburg How far do you travel during that time

period?

you pass milepost 326 near Valley Forge, then milepost

351 near Philadelphia How far do you travel during that

time period?

January temperatures (in degrees Fahrenheit) for a

city are given Find the distance between the numbers

to determine the range of temperatures for January

81 McGrath, Alaska: lowest:

highest:

82 Flagstaff, Arizona: lowest:

highest:

83– 88, the accounting department of an Internetstart-up company is checking to see whether variousactual expenses differ from the budgeted expenses bymore than $500 or by more than 5% Complete themissing parts of the table Then determine whetherthe actual expense passes the “budget variance test.”

89–94, the quality control inspector for a tire factory istesting the rim diameters of various tires A tire

is rejected if its rim diameter varies too much from itsexpected measure The diameter should not differ bymore than 0.02 inch or by more than 0.12% of theexpected diameter measure Complete the missingparts of the table Then determine whether the tire ispassed or rejected according to the inspector’s guidelines

Expected Actual Diameter, b Diameter, a 0.0012b

(a) Are the values of the expressions always equal? If not,under what conditions are they unequal?

(b) If the two expressions are not equal for certain values

of u and v, is one of the expressions always greater than

the other? Explain

that a real number is positive and saying that a real number

is nonnegative? Explain

97 Describe the differences among the sets of natural

num-bers, integers, rational numnum-bers, and irrational numbers

real number a?Explain ⱍaⱍ a

3 2

Section 0.2

The Basic Rules

of Algebra

Algebraic Expression

A collection of letters (called variables) and real numbers (called constants)

that are combined using the operations of addition, subtraction,

multiplica-tion, and division is an algebraic expression (Other operations can also be

used to form an algebraic expression.)

■ Identify the terms of an algebraic expression

■ Evaluate an algebraic expression

■ Identify basic rules of algebra

■ Perform operations on real numbers

■ Use a calculator to evaluate an algebraic expression

Algebraic Expressions

One of the basic characteristics of algebra is the use of letters (or combinations

of letters) to represent numbers The letters used to represent numbers are called

variables, and combinations of letters and numbers are called algebraic expressions Some examples of algebraic expressions are

and

The terms of an algebraic expression are those parts that are separated by

addition For example, the algebraic expression has three terms: and 8 Note that rather than is a term, because

The terms and are the variable terms of the expression, and 8 is the

constant term of the expression The numerical factor of a variable term is the coefficient of the variable term For instance, the coefficient of the variable term

is and the coefficient of the variable term is 1.

Example 1 Identifying the Terms of an Algebraic Expression

Algebraic Expression Terms

To evaluate the expression

for the

values 2 and 5, use the last

entry feature of a graphing

utility.

1 Evaluate

2 Press [ENTRY] (recalls

previous expression to the

home screen).

3 Cursor to 2, replace 2 with

For specific keystrokes for the

last entry feature, go to the text

When you evaluate an

expres-sion with grouping symbols

The Substitution Principle states, “If then can be replaced by

in any expression involving ” You use this principle to evaluate an

algebraic expression by substituting numerical values for each of the variables in the expression In the first evaluation shown below, 3 is substituted for in the expression

Expression Variable Substitution Expression

Undefined

Example 3 Evaluating Algebraic Expressions

Evaluate each algebraic expression when and

SOLUTION

a When and the expression has a value of

b When the expression has a value of

c When the expression has a value of

Basic Rules of Algebra The four basic arithmetic operations are addition, multiplication, subtraction, and division, denoted by the symbols or and respectively Of these, addition and multiplication are considered to be the two primary arithmetic oper- ations Subtraction and division are defined as the inverse operations of addition and multiplication, as follows.

The symbol indicates when to use graphing technology or a symbolic computer algebra system

to solve a problem or an exercise The solutions of other exercises may also be facilitated by use ofappropriate technology

Subtraction: Add the opposite Division: Multiply by the reciprocal

If then

In these definitions, is called the additive inverse (or opposite) of b, and

is called the multiplicative inverse (or reciprocal) of b In place of you can use the fraction symbol In this fractional form, is called the numera-

tor of the fraction and b is called the denominator.

The basic rules of algebra, listed below, are true for variables and algebraic

expressions as well as for real numbers.

Because subtraction is defined as “adding the opposite,” the Distributive Property is also true for subtraction For instance, the “subtraction form” of

is a 共b  c兲  a关b  共c兲兴  ab  a共c兲  ab  ac.

Basic Rules of Algebra

Let and be real numbers, variables, or algebraic expressions.

Commutative Property of Addition

Commutative Property of Multiplication

Associative Property of Addition

Associative Property of Multiplication

Distributive Property

Additive Identity Property

Multiplicative Identity Property

Additive Inverse Property

Multiplicative Inverse Property

a, b,

Example 4 Identifying the Basic Rules of Algebra

Identify the rule of algebra illustrated by each statement.

a This equation illustrates the Commutative Property of Multiplication.

b This equation illustrates the Additive Inverse Property.

c This equation illustrates the Associative Property of Addition

d This equation illustrates the Distributive Property in reverse order.

Distributive Property

e This equation illustrates the Multiplicative Inverse Property Note that it is

important that be a nonzero number If were allowed to be zero, you would

be in trouble because the reciprocal of zero is undefined.

Identify the rule of algebra illustrated by each statement.

a.

The following three lists summarize the basic properties of negation, zero,

and fractions When you encounter such lists, you should not only memorize a verbal description of each property, but you should also try to gain an intuitive feeling for the validity of each.

x2 5  5  x2

3x2 1  3x2

x x

Be sure you see the difference between the opposite of a number and a negative number If is negative, then its opposite, is positive For instance,

The “or” in the Zero-Factor Property includes the possibility that both

factors are zero This is called an inclusive or, and it is the way the word “or” is

always used in mathematics.

Properties of Zero

Let a and b be real numbers, variables, or algebraic expressions Then the

following properties are true.

In Property 1 (equivalent fractions) the phrase “if and only if” implies two

is: If where ad  bc, b  0 and d a 兾b  c兾d,  0, then a 兾b  c兾d ad  bc.

Properties of Fractions

Let and be real numbers, variables, or algebraic expressions such

1 Equivalent fractions: if and only if

2 Rules of signs: and

3 Generate equivalent fractions:

4 Add or subtract with like denominators:

5 Add or subtract with unlike denominators:

Example 5 Properties of Zero and Properties of Fractions

divisors of For example, 2 and 3 are factors of 6 because A prime

number is a positive integer that has exactly two factors: itself and 1 For

example, 2, 3, 5, 7, and 11 are prime numbers, whereas 1, 4, 6, 8, 9, and 10 are

not The numbers 4, 6, 8, 9, and 10 are composite because they can be written as

the products of two or more prime numbers The number 1 is neither prime nor

composite The Fundamental Theorem of Arithmetic states that every positive

integer greater than 1 is a prime number or can be written as the product of prime

numbers in precisely one way (disregarding order) For instance, the prime factorization of 24 is

When you are adding or subtracting fractions that have unlike denominators, you can use Property 4 of fractions by rewriting both of the fractions so that they

have the same denominator This is called the least common denominator

method.

Example 6 Adding and Subtracting Fractions

Evaluate

denominator (LCD) Use the LCD, 45, to rewrite the fractions and simplify.

ab  c, c

b, a,

The LCD is the product of the

prime factors, with each factor

given the highest power of its

occurrence in any denominator.

An equation is a statement of equality between two expressions So, the statement

instance, because and both represent the number 5, you can write

Three important properties of equality follow.

In algebra, you often rewrite expressions by making substitutions that are permitted under the Substitution Principle Two important consequences of the Substitution Principle are the following rules.

The first rule allows you to add the same number to each side of an equation The second allows you to multiply each side of an equation by the same number The converses of these two rules are also true and are listed below

So, you can also subtract the same number from each side of an equation as well

as divide each side of an equation by the same nonzero number.

Calculators and Rounding

The table below shows keystrokes for several similar functions on a standard scientific calculator and a graphing calculator These keystrokes may not be the same as those for your calculator Consult your user’s guide for specific keystrokes.

Graphing Calculator Scientific Calculator

For example, you can evaluate on a graphing calculator or a scientific calculator as follows.

Graphing Calculator Scientific Calculator

Example 7 Using a Calculator

When rounding decimals, look at the decision digit (the digit at the right of

the last digit you want to keep) Round up when the decision digit is 5 or greater, and round down when it is 4 or less.

Rounded to Number Three Decimal Places

7  共5 3兲

14.8ⴝ

ⴛ37% of 40

9ⴝ

ⴜ

冈

x 2 ⴚ

冇

ⴛ3共10  42兲  2

ⴝ

y x ⴜ

24  23

244ⴝ

ⴚ ⴙⲐⴚ

x 2

122 100

8ⴝ

ⴛ ⴚ

7  共5 3兲

Be sure you see the

differ-ence between the change

subtraction key , as used in

Use a calculator to evaluate

Then round the result to two

3 Is the expression always negative? Explain

4 Explain how to divide by when and are nonzero real numbers

d

b, c,

c/d

a/b ⴚx

c b, a,

冇a ⴚ b冈1 c ⴝ a ⴚ冇b 1 c冈

C O N C E P T C H E C K

ⴚ

In Exercises 1–6, identify the terms of the algebraic

In Exercises 11–16, evaluate the expression for each

value of (If not possible, state the reason.)

3 8

共2 3兲 5

8 3 4 2

5 7 8

10

11 6

3313 66 5

81

45 6

27 354

The following warm-up exercises involve skills that were covered in earlier sections You will usethese skills in the exercise set for this section For additional help, review Section 0.1

In Exercises 1– 4, place the correct inequality symbol or between the

two real numbers

In Exercises 51–54, use a calculator to evaluate the

expression (Round to two decimal places.)

shows the types of expenses for the federal government in

2005 (Source: Office of Management and Budget)

(a) What percent of the total expenses was the amount

spent on Social Security?

(b) The total of the 2005 expenses was $2,472,200,000,000

Find the amount spent for each category in the circle

graph (Round to the nearest billion dollars.)

study that have a particular health risk is 39.5% The total

number of people in the study is 12,857 How many people

have the health risk?

cancer treatment showing a decrease in tumor size is

49.2% There are 3445 patients in the trial How many

patients show a decrease in tumor size?

that corresponds to each set of keystrokes

evaluate each algebraic expression on either a scientific or

a graphing calculator

In Exercises 64 and 65, a breakdown of pet spendingfor one year in the United States is given Find the percent of total pet spending for each subcategory.Then use a spreadsheet software program to make alabeled circle graph for the percent data (Source: American Pet Products Manufacturers Association)

64 Total pet spending (2005): $36.3 billion

Supplies/OTC medicine: $8.7 billionLive animal purchases: $1.7 billionGrooming and boarding: $2.5 billion

65 Total pet spending (2006): $38.4 billion

Supplies/OTC medicine: $9.3 billionLive animal purchases: $1.8 billionGrooming and boarding: $2.7 billion

Tim Sloan /AFP/Getty Images

PetSmart, the largest U.S pet store chainwith 909 stores, has grown by offering petlodging services in some stores PetsHotels provide amenities such as supervised play areaswith toys and slides, hypoallergenic lambskinblankets, TV, healthy pet snacks, and special fee services such as grooming, training, andphoning pet parents These services are twice asprofitable as retail sales, and they tend toattract greater sales as well PetSmart’s saleswere 29% higher in stores with establishedPetsHotels than in those without them In 2006,PetSmart had a goal to expand from 62 to 435PetsHotels

Internet, or some other reference source to findinformation about “special services” companiesexperiencing strong growth as in the example above.Write a brief report about one of these companies.The symbol indicates an exercise in which you are instructed

to use a spreadsheet

■ Use properties of exponents.

■ Use scientific notation to represent real numbers

■ Use a calculator to raise a number to a power

■ Use interest formulas to solve an application problem

Properties of Exponents Repeated multiplication of a real number by itself can be written in exponential

form Here are some examples.

Repeated Multiplication Exponential Form

When multiplying exponential expressions with the same base, add

exponents.

Add exponents when multiplying

For instance, to multiply and you can write

On the other hand, when dividing exponential expressions, subtract exponents.

That is,

Subtract exponents when dividing

These and other properties of exponents are summarized in the list on the following page.

an a a a a

n a

S T U D Y T I P

It is important to recognize the

difference between exponential

In the parentheses

indi-cate that the exponent applies to

the negative sign as well as to

exponent applies only to the 2.

Similarly, in the

parenthe-ses indicate that the exponent

applies to the 5 as well as to the

Notice that these properties of exponents apply for all integers m and n, not

just positive integers For instance, by the Quotient of Powers Property,

Example 1 Using Properties of Exponents

Let and be real numbers, variables, or algebraic expressions, and let and be integers (Assume all

denominators and bases are nonzero.)

a

The next example shows how expressions involving negative exponents can

be rewritten using positive exponents.

Example 2 Rewriting with Positive Exponents

Product of Powers Property

Power of a Power PropertyDefinition of negative exponentSimplify

Example 3 Ratio of Volume to Surface Area

The volume and surface area of a sphere are given by

and where is the radius of the sphere A spherical weather balloon has a radius of

2 feet, as shown in Figure 0.7 Find the ratio of the volume to the surface area.

4  22 1

3 共2兲  2 3

V 兾S r

S  4 r2

V  4

3 r3

S V

x1 1

x

S T U D Y T I P

Rarely in algebra is there only

one way to solve a problem.

Don’t be concerned if the steps

you use to solve a problem are

not exactly the same as the steps

presented here The important

thing is to use steps that you

understand and that, of course,

are justified by the rules of

alge-bra For instance, you might

prefer the following steps to