Brief Calculus with Applications / Edition 2

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Overview

This book, modern in its writing style as well as in its applications, contains numerous exercises—both skill oriented and applications—, real data problems, and a problem solving method. The book features exercises based on data form the World Wide Web, technology options for those who wish to use a graphing calculator, review boxes, strategic checkpoints, interactive activities, section summaries and projects, and chapter openers and reviews. For anyone who wants to see and understand how mathematics are used in everyday life.

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Product Details

  • ISBN-13: 9780130655912
  • Publisher: Prentice Hall
  • Publication date: 9/28/2002
  • Edition description: 2ND
  • Edition number: 2
  • Pages: 784
  • Product dimensions: 8.28 (w) x 10.36 (h) x 1.39 (d)

Meet the Author

Bill Armstrong. Bill is a native of Ohio and became a hard-core Buckeyes fan after earning both his Bachelors and Masters degrees in Mathematics at The Ohio State University.


Bill taught at numerous colleges including Ohio State and Phoenix College, before taking his current position at Lakeland Community College near Cleveland, Ohio. Bill enjoys working with students at all levels and teaches courses ranging from Algebra to Differential Equations. He employs various teaching strategies to interest and motivate students including using humor to lighten the subject matter and inviting comments from students. His enjoyment of teaching and constant interaction with students has earned him a reputation as an innovative, enthusiastic, and effective teacher.


When Bill is not teaching, tutoring during office hours, or writing, he enjoys playing pool and golf, coaching his sons' little league baseball teams, and hanging out at home with his wife... not necessarily in that order!


Don Davis. Also a native of Ohio, Don earned a Bachelor of Science degree in Education, specializing in Political Science, Economics, and Mathematics from Bowling Green State University in Bowling Green, Ohio. After teaching at the high school level, Don received his Master of Science degree in Mathematics from Ohio University. He taught at Ohio State University - Newark before joining the Lakeland Community College faculty. With his background in Economics, Don is always searching for ways to apply mathematics to the "real world" and to connect mathematical concepts to the various courses his students are taking. By using technology, alongwith interesting, practical applications, Don brings his many mathematics classes to life for students. Like Bill, one typically finds Don in his office helping students understand the concepts of his courses. His dedication to his students makes him a sought-after teacher.


In his free time, Don is often found playing with his three children or scurrying after one of the many pets currently in residence at his home including a rat, two parakeets, a turtle, a lizard, a dog, and two cats. Occasionally, he enjoys a quiet night at home.

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Table of Contents

1. Functions, Modeling and Average Rate of Change.
Coordinate Systems and Functions. Introduction to Problem Solving. Linear Functions and Average Rate of Change. Quadratic Functions and Average Rate of Change on an Interval. Operations on Functions. Rational, Radical and Power Functions. Exponential Functions. Logarithmic Functions. Regression and Mathematical Models (Optional Section).

2. Limits, Instantaneous Rate of Change and the Derivative.
Limits. Limits and Asymptotes. Problem Solving: Rates of Change. The Derivative. Derivatives of Constants, Powers and Sums. Derivatives of Products and Quotients. Continuity and Nondifferentiability.

3. Applications of the Derivative.
The Differential and Linear Approximations. Marginal Analysis. Measuring Rates and Errors.

4. Additional Differentiation Techniques.
The Chain Rule. Derivatives Logarithmic Functions. Derivatives of Exponential Functions. Implicit Differentiation and Related Rates. Elasticity of Demand.

5. Further Applications of the Derivative.
First Derivatives and Graphs. Second Derivatives and Graphs. Graphical Analysis and Curve Sketching. Optimizing Functions on a Closed Interval. The Second Derivative Test and Optimization.

6. Integral Calculus.
The Indefinite Integral. Area and the Definite Integral. Fundamental Theorem of Calculus. ProblemSolving: Integral Calculus and Total Accumulation. Integration by u-substitution. Integrals That Yield Logarithmic and Exponential Functions. Differential Equations: Separation of Variables. Differential Equations: Growth and Decay.

7. Applications of Integral Calculus.
Average Value of a Function and the Definite Integral in Finance. Area Between Curves and Applications. Economic Applications of Area between Curves. Integration by Parts. Numerical Integration. Improper Integrals.

8. Calculus of Several Variables.
Functions of Several Independent Variables. Level Curves, Contour Maps and Cross-Sectional Analysis. Partial Derivatives and Second-Order Partial Derivatives. Maxima and Minima. Lagrange Multipliers. Double Integrals.

Appendix A. Essentials of Algebra.
Appendix B. Calculator Programs.
Appendix C. Selected Proofs.
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Preface

Audience

In preparing to write this text, we talked with many colleagues who teach a brief or applied math course to find out if they experienced the same difficulties in teaching this course_ that we have encountered. What we learned is that while there is some similarity in the topics covered and in how much time is spent on each area, there is remarkable uniformity in the needs of students who enroll in these diverse courses. Professors at Community Colleges, Universities, and Liberal Arts Colleges all told us that their students are generally unmotivated, unsure of their algebra skills, uncomfortable with translating English into mathematics, and unschooled in how to set up problems for solution. Armed with this knowledge, we prepared the second edition of Brief Calculus to address these fundamental needs.

As with other applied calculus textbooks, our text may be used in either a one or two term course for students majoring in economics, business, or social or behavioral sciences. We have organized the topics for maximum flexibility so that the text may be adapted to any college or university's curriculum. However, that is where the similarity ends. We have crafted this book around five key principles designed to address students' needs:

  • Present the mathematics in language that students can read and understand
  • Teach good problem solving techniques and provide ample practice
  • Use real data applications to keep it interesting
  • Provide timely reinforcement of algebra and other essential skills
  • Let instructors decide whether to incorporate technology

Present the Mathematics in LanguageThat Students Can Read and Understand

By writing this text in a conversational, easy-to-read style, we strive to evoke the one-on-one communication of a tutorial session. When students find that they can understand the clear presentation and follow the interesting, real world examples, we believe they will get into the habit of reading the text. Although we have written a text that is accessible, we have been careful not to sacrifice the proper depth of coverage and necessary rigor required of applied calculus. We are confident both objectives have been met.

Teach Good Problem Solving Techniques and Provide Ample Practice

Problem Solving Sections

A new feature of the Second Edition is the addition of dedicated problem solving sections. Before explaining how to solve an entire class of problems, we provide a special section that demonstrates how to apply the appropriate mathematical tools to analyze a given type of problem. For example Section 1.2, Introduction to Problem Solving, introduces our general approach to problem solving and then explores mathematical models and their properties, and how numerical solutions to mathematical models are interpreted.

  • Section 2.3—Problem Solving: Rates of Change connects the average rate of change and secant line slope to instantaneous rate of change and tangent line slope using numerical and graphical techniques.
  • Section 6.4—Problem Solving: Integral Calculus and Total Accumulation explains how the definite integral can be used, given a rate of change function, to determine a continuous sum.

Many of the exercises in these problem solving sections prepare students for subsequent sections because they introduce exercises that are solved later in the textbook.

Problem Solving Method

New to the second edition, our clearly developed problem solving method is the single most distinctive, and user-friendly feature of the book. Frequently, applied mathematics instructors hear students comment that "I don't even know how to begin this problem." or "If the problem was just set up for me, I could solve it." Because skills such as setting up problems and writing the solution in its proper context can be a major challenge for applied calculus students, we have integrated a problem solving method throughout the text. The steps of the problem solving method are referenced throughout the text. We use the phrases "Understand the Situation" and "Interpret the Solution" to identify two critical steps in problem solving.

  • When solving an application example, we first need to Understand the Situation. In these clearly labeled paragraphs we state the quantity we are trying to determine and identify the method used to determine the solution. Frequently, we reveal the thought processes necessary to attack the problem.
  • After determining a numerical solution in a traditional solution step, we Interpret the Solution. In the interpretation step, we write a concluding sentence that conveys the meaning of the numerical answer in the context of the application. To reinforce these techniques, we often ask students to solve and interpret the solution to various applied problems in the exercise sets.

Exercises

The comprehensive exercise sets are the heart of our textbook. The typical exercise set contains numerous skill builder problems, a generous selection of applications from many different disciplines. Another lesson learned from our market research is that many applied calculus textbooks fail to provide enough exercises for the student to grasp the course content. With greater than 3500 exercises, we are confident our textbook has more than enough exercises to meet student's needs.

Use Real Data Applications to Keep It Interesting

Students in this course tend to be very pragmatic; they want to know why they must learn the mathematical content in this course. Including many real data modeling applications in examples and exercises helps to answer their unstated question and provides motivation and interest. Many of the models, parameters, and scenarios in the examples and exercises are based on data gathered from the U.S. Statistical Abstract, the Census Bureau, and other reliable sources, for which URLs are always given. In the second edition, we have updated hundreds of mathematical models based on the published results of the 2000 census. For example, we include real data modeling applications that examine the rate of change in U.S. corporate profits after taxes (Source: U.S. Bureau of Economic Analysis, bea.doc.gov) in the integral calculus chapters. Other examples of real data applications include:

  • The number of milligrams of cholesterol consumed each day per person in the U.S. (Source: U.S. Department of Agriculture, usda.gov).
  • The annual total expenditures on pollution abatement in the U.S. (Source: U.S. Statistical Abstract, census.gov/statab/www).
  • The rate of change in the number of arrests for drug abuse violations by American adults (Source: U.S. Census Bureau, census.gov).

The quantity, quality, and variation of these types of applications are simply not found in other applied calculus textbooks. We believe that real data modeling applications not only keep the course content relevant and fresh, but compel students to interpret the numerical solution in the context of the problem they have solved.

Provide Timely Reinforcement of Algebra and Other Essential Skills

One of the major challenges faced by students, and frustrations encountered by instructors, is weak preparation in algebra and other essential skills. Even students who have proficient algebra skills are often rusty and unsure of which algebraic tool to apply. Further, the content demands of an applied calculus course do not allow for extensive time spent on review. In an effort to address this pervasive problem, we have developed the "From Your Toolbox" feature.

From Your Toolbox

When appropriate, the "From Your Toolbox" feature directs students to read background material in the Algebra Review or other appendices. In addition, this feature is used to review previously introduced definitions, theorems, or properties as needed. By providing a brief review when it is needed, students stay on task and do not need to flip back to hunt through previous sections for key information.

Let Instructors Decide Whether to Incorporate Technology

As graduates and instructors of The Ohio State University, we began using graphing calculator technology in the classroom long before it was fashionable to do. Based on our years of experience in this area, we have seen the strengths of using a graphing calculator and the drawbacks as well. Our philosophy is to let the instructor, rather than the textbook, determine how much or how little graphing calculators, spreadsheets, or other desktop applications are used in the classroom. Consequently, we have developed the Technology Option in our text, which allows each instructor to decide whether or not to use technology in the curriculum.

Technology Option

These shaded, optional parts are easy to find, or to skip, and typically follow examples. The content of these parts mirrors the traditional presentation but shows how the answer to a particular example may be found using a graphing calculator. Although keystroke commands are not given, we provide answers to general questions students may have. All screen shots included in the text are from the Texas Instruments TI-83 calculator. Keystrokes and commands for various models and brands of calculator are found at the online graphing calculator manual found at the companion website at . This site is designed to supplement the textbook by offering a variety of teaching and learning resources for each chapter. This includes a net search of topics relevant to the chapter, links to related chapter topics, solutions to selected interactive activities, quizzes, program downloads for various models of graphing calculators, and an online graphing calculator reference manual.

TestGen-ft for Windows and Macintosh (ISBN 0-13-067417-6). This algorithmic software-testing program allows instructors to create tests quickly and efficiently using the supplied questions or to personalize tests using the built-in editing features.

Test Item File (ISBN 0-13-067415-X). Hard copy of the algorithmic computerized testing materials provides a quick reference for the testing software.

Read More Show Less

Introduction

Audience

In preparing to write this text, we talked with many colleagues who teach a brief or applied math course to find out if they experienced the same difficulties in teaching this course_ that we have encountered. What we learned is that while there is some similarity in the topics covered and in how much time is spent on each area, there is remarkable uniformity in the needs of students who enroll in these diverse courses. Professors at Community Colleges, Universities, and Liberal Arts Colleges all told us that their students are generally unmotivated, unsure of their algebra skills, uncomfortable with translating English into mathematics, and unschooled in how to set up problems for solution. Armed with this knowledge, we prepared the second edition of Brief Calculus to address these fundamental needs.

As with other applied calculus textbooks, our text may be used in either a one or two term course for students majoring in economics, business, or social or behavioral sciences. We have organized the topics for maximum flexibility so that the text may be adapted to any college or university's curriculum. However, that is where the similarity ends. We have crafted this book around five key principles designed to address students' needs:

  • Present the mathematics in language that students can read and understand
  • Teach good problem solving techniques and provide ample practice
  • Use real data applications to keep it interesting
  • Provide timely reinforcement of algebra and other essential skills
  • Let instructors decide whether to incorporate technology

Present the Mathematics in Language That Students CanRead and Understand

By writing this text in a conversational, easy-to-read style, we strive to evoke the one-on-one communication of a tutorial session. When students find that they can understand the clear presentation and follow the interesting, real world examples, we believe they will get into the habit of reading the text. Although we have written a text that is accessible, we have been careful not to sacrifice the proper depth of coverage and necessary rigor required of applied calculus. We are confident both objectives have been met.

Teach Good Problem Solving Techniques and Provide Ample Practice

Problem Solving Sections

A new feature of the Second Edition is the addition of dedicated problem solving sections. Before explaining how to solve an entire class of problems, we provide a special section that demonstrates how to apply the appropriate mathematical tools to analyze a given type of problem. For example Section 1.2, Introduction to Problem Solving, introduces our general approach to problem solving and then explores mathematical models and their properties, and how numerical solutions to mathematical models are interpreted.

  • Section 2.3—Problem Solving: Rates of Change connects the average rate of change and secant line slope to instantaneous rate of change and tangent line slope using numerical and graphical techniques.
  • Section 6.4—Problem Solving: Integral Calculus and Total Accumulation explains how the definite integral can be used, given a rate of change function, to determine a continuous sum.

Many of the exercises in these problem solving sections prepare students for subsequent sections because they introduce exercises that are solved later in the textbook.

Problem Solving Method

New to the second edition, our clearly developed problem solving method is the single most distinctive, and user-friendly feature of the book. Frequently, applied mathematics instructors hear students comment that "I don't even know how to begin this problem." or "If the problem was just set up for me, I could solve it." Because skills such as setting up problems and writing the solution in its proper context can be a major challenge for applied calculus students, we have integrated a problem solving method throughout the text. The steps of the problem solving method are referenced throughout the text. We use the phrases "Understand the Situation" and "Interpret the Solution" to identify two critical steps in problem solving.

  • When solving an application example, we first need to Understand the Situation. In these clearly labeled paragraphs we state the quantity we are trying to determine and identify the method used to determine the solution. Frequently, we reveal the thought processes necessary to attack the problem.
  • After determining a numerical solution in a traditional solution step, we Interpret the Solution. In the interpretation step, we write a concluding sentence that conveys the meaning of the numerical answer in the context of the application. To reinforce these techniques, we often ask students to solve and interpret the solution to various applied problems in the exercise sets.

Exercises

The comprehensive exercise sets are the heart of our textbook. The typical exercise set contains numerous skill builder problems, a generous selection of applications from many different disciplines. Another lesson learned from our market research is that many applied calculus textbooks fail to provide enough exercises for the student to grasp the course content. With greater than 3500 exercises, we are confident our textbook has more than enough exercises to meet student's needs.

Use Real Data Applications to Keep It Interesting

Students in this course tend to be very pragmatic; they want to know why they must learn the mathematical content in this course. Including many real data modeling applications in examples and exercises helps to answer their unstated question and provides motivation and interest. Many of the models, parameters, and scenarios in the examples and exercises are based on data gathered from the U.S. Statistical Abstract, the Census Bureau, and other reliable sources, for which URLs are always given. In the second edition, we have updated hundreds of mathematical models based on the published results of the 2000 census. For example, we include real data modeling applications that examine the rate of change in U.S. corporate profits after taxes (Source: U.S. Bureau of Economic Analysis, www.bea.doc.gov) in the integral calculus chapters. Other examples of real data applications include:

  • The number of milligrams of cholesterol consumed each day per person in the U.S. (Source: U.S. Department of Agriculture, www.usda.gov).
  • The annual total expenditures on pollution abatement in the U.S. (Source: U.S. Statistical Abstract, www.census.gov/statab/www).
  • The rate of change in the number of arrests for drug abuse violations by American adults (Source: U.S. Census Bureau, www.census.gov).

The quantity, quality, and variation of these types of applications are simply not found in other applied calculus textbooks. We believe that real data modeling applications not only keep the course content relevant and fresh, but compel students to interpret the numerical solution in the context of the problem they have solved.

Provide Timely Reinforcement of Algebra and Other Essential Skills

One of the major challenges faced by students, and frustrations encountered by instructors, is weak preparation in algebra and other essential skills. Even students who have proficient algebra skills are often rusty and unsure of which algebraic tool to apply. Further, the content demands of an applied calculus course do not allow for extensive time spent on review. In an effort to address this pervasive problem, we have developed the "From Your Toolbox" feature.

From Your Toolbox

When appropriate, the "From Your Toolbox" feature directs students to read background material in the Algebra Review or other appendices. In addition, this feature is used to review previously introduced definitions, theorems, or properties as needed. By providing a brief review when it is needed, students stay on task and do not need to flip back to hunt through previous sections for key information.

Let Instructors Decide Whether to Incorporate Technology

As graduates and instructors of The Ohio State University, we began using graphing calculator technology in the classroom long before it was fashionable to do. Based on our years of experience in this area, we have seen the strengths of using a graphing calculator and the drawbacks as well. Our philosophy is to let the instructor, rather than the textbook, determine how much or how little graphing calculators, spreadsheets, or other desktop applications are used in the classroom. Consequently, we have developed the Technology Option in our text, which allows each instructor to decide whether or not to use technology in the curriculum.

Technology Option

These shaded, optional parts are easy to find, or to skip, and typically follow examples. The content of these parts mirrors the traditional presentation but shows how the answer to a particular example may be found using a graphing calculator. Although keystroke commands are not given, we provide answers to general questions students may have. All screen shots included in the text are from the Texas Instruments TI-83 calculator. Keystrokes and commands for various models and brands of calculator are found at the online graphing calculator manual found at the companion website.

Exercises That Assume Technology

Exercises that assume the use of a graphing calculator are clearly marked with a symbol so they can be assigned or skipped as desired by the instructor. We recognize that the graphing calculator is simply a tool to be used in the understanding of mathematics. We have been very careful to introduce the technology only where it is appropriate and not to let its use overshadow the mathematics.

Content Features and Highlights

Rate of Change Theme. Because we believe it is important for students to understand that calculus is the study of rates of change, we have highlighted this theme throughout the text. Beginning in Chapter 1, the basic algebraic and transcendental functions are reviewed in a concise and comprehensive manner. As each type of function is introduced, the average rate of change of the function on a closed interval is presented in examples and exercises. From the onset, appropriate units and an emphasis on interpreting, rather than merely finding a numerical answer, are stressed.

  • For example, the solution to an exercise in Section 1.4 reads, "From 1980 to 1990, the number of lawyers in private practice increased at an average rate of 17.07 thousand per year." This requires students to determine the average rate of change of a nonlinear model on a closed interval. We have found that by introducing the difference quotient in this manner early and often in Chapter 1 we create a smooth transition to computing the instantaneous rate of change and the derivative in Chapter 2.
  • The remaining Chapters continue to highlight the rate of change theme, along with the importance of proper units and a sound interpretation of the solution. If a student who uses Brief Calculus: Solving Problems in Business, Economics, and the Social and Behavioral Sciences is asked, "What is calculus?" we are confident that the student will proclaim, "It is the study of rates of change!"

Use of the Differential

To supplement the rate of change theme, we have paid particular attention to the use of the differential in applied calculus topics. The differential is introduced early in Chapter 3 and is used as a mathematical tool to introduce new topics in later sections.

  • In Section 3.2 for example, marginal analysis is introduced by means of the differential and then we show that the marginal business functions are a type of differential.
  • In Section 3.3, relative errors and the relative rate of change are explained by the Use of the differential.
  • In Chapter 4, he differential is used to motivate the formula for elasticity of demand.
  • In Chapter 6, we revisit the differential again when we introduce integration by u-substitution. In contrast to many applied calculus texts which teach the differential as an isolated topic, we present it as a powerful tool that is useful in both differential and integral calculus.

The Definite Integral as a Continuous Sum

We have found that many of the applications of the definite integral in applied calculus textbooks tend to be contrived, esoteric, and difficult to interpret. To address this shortcoming, we have made the use of the definite integral to compute a total accumulation over a continuous interval. Introduced in the Chapter 6 problem solving section, students learn that integrating a rate function (that is, a derivative), on an interval gives the total accumulation of the dependent variable values on that interval. For example, the total difference in cost C(b) - C(a) can be determined by integrating the marginal cost function MC(x) over the interval a,b. This central idea is used throughout the integral calculus unit, resulting in applications that are richer and more interesting than are found in most texts.

Chapter Features

Chapter Openers. The first page of each chapter lists the sections included in that chapter, along with a photo and representative graphs or figures that foreshadow the fundamental ideas presented in the chapter in the context of an application. "What We Know" reiterates what information has been learned in previous chapters and "Where Do We Go" explains what topics will be covered. The chapter opener helps create a roadmap to guide the student through the book and underscore the connections between topics.

Flashbacks. Selected examples used earlier in the textbook are revisited in the Flashback feature. The Flashback carries over an applied example from a previous section and then extends the content of the example by considering new questions. In this manner, new topics are introduced in a more natural way within a familiar context. Moreover, the Flashback often reviews the necessary skills and concepts from previous chapters. We believe that this pedagogical technique of using applications previously discussed allows students to concentrate on new topics using familiar applications.

Interactive Activities. Extensions to completed examples are included in the Interactive Activities that appear after many examples. The Interactive Activities may ask students to solve a problem using a different method, to discover a pattern that can lead to a mathematical property, or to explore additional properties of recently introduced topics. Interactive Activities may be used in a number of ways in the classroom. Instructors may assign them as critical thinking exercises, or they may be used as a springboard for classroom discussion, or they may provide a vehicle for collaborative activities. Interactive Activities that include the Web icon have solutions provided on the companion website.

Checkpoints. At strategic points in each section, an example is followed by a Checkpoint. Each Checkpoint directs a student to complete a selected odd numbered problem in that section's exercise set. Guiding the student to a parallel exercise to-the example problem helps to reinforce the topic at hand and ensures that the recently introduced skill or concept is practiced immediately. This pedagogical tool promotes interaction between the text and the student and helps students to develop good study habits. Students who use the Checkpoints will quickly learn to take ownership of the course material.

Notes. Another popular feature is the Notes that appear after many definitions, theorems, and properties. Notes are used to clarify a mathematical idea verbally and to provide students with additional insight into the material. Many times these Notes echo what a professor might state in the classroom to help the student understand the definition, theorem, or property.

Section Projects. At the end of each exercise set, a Section Project presents a series of questions that ask students to explore the idea that is presented. Some of the projects are based on real data and ask students to use the regression capabilities of their calculator to determine a model. Others give a step-by-step procedure that can be used to solve classic problems in applied calculus. Instructors can use section projects as standard hand-in assignments, collaborative activities, or for demos in graphing calculator usage.

Section Summary. At the end of each section, a short summary highlights the important concepts of the section. Section summaries are a convenient resource that students may use while completing exercises, or as a springboard for test review.

End of Chapter Material. Each chapter concludes with three components; first is a summary called Why We Learned It. This narrative outlines the major topics of each chapter and describes how the topics are used in various careers. An extensive set of Chapter Review Exercises is designed to augment the exercises in each section of the chapter. Concluding the chapter is the Chapter Project. These extensive exercises use real data modeling to explore and extend the ideas discussed in each chapter. They may be assigned as individual homework or as a group project.

Supplements

Instructor's Solution Manual (ISBN 0-13-067412-5). Written by Denyse Kerr and checked for accuracy by the authors and John Cruthirds (North Georgia College and State University), this volume contains complete solutions to all of the exercises, and the section and chapter projects.

Student's Solution Manual (ISBN 0-13-067413-3). Written by Denyse Kerr, and checked for accuracy by the authors and John Cruthirds (North Georgia College and State University), this volume contains complete solutions to all of the odd numbered section and review exercises in the textbook.

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