Brief Calculus: The Study of Rates of Change Updated Edition

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9780131019874: Brief Calculus: The Study of Rates of Change Updated Edition

This is the first post-reform calculus book that emphasizes applications and innovative approaches while preserving the underlying mathematics. Rich and varied, the abundant applications that use real data motivate the mathematics and underscore the importance of mathematical underpinnings. The graphing calculator and applications that use data on the World Wide Web are integrated as optional resources to support mathematical ideas, techniques, and applications. The informal tone and abundance and variety of elementary and moderately difficult problems make this book the ideal choice for math anxious people. Chapter 1 reviews functions and provides a solid foundation for the idea that the average rate of change over an interval is equivalent to the slope of the secant line. Introduction of limits at infinity and infinite limits. Strong introduction of the differential. This book devotes two sections to the difficult concept of optimization. For those working in a non-math field, who need/want to learn calculus.

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Audience and Prerequisites

Brief Calculus: The Study of Rates of Change is intended for a one or two-term calculus course for students majoring in business, economics, social or life sciences. Students who have completed an appropriate college algebra or precalculus course are prepared to study the topics presented in this textbook. The text is designed for both two-year and four-year schools.

Style and Approach

In contrast to many brief and applied calculus textbooks on the market, our book is written for the student. Our conversational writing style explains the mathematics clearly and simply and employs real data from the Internet to motivate and underscore the relevance of calculus in a wide range of applications. The book draws from the best ideas of reform mathematics, yet is built on a tradition of solid mathematics. As reviewer Rene Barrientos from Miami-Dade Community College put it, "This is the first post-reform calculus text that emphasizes applications without giving up the symbolic manipulations and coverage required of a rigorous calculus course."

The core concepts of calculus are introduced in applied settings using the concept commonly known as the Rule of Three (numerical, graphical, and algebraic). The Rule of Three is one tool we use to aid students' understanding of new ideas. The exploration of concepts and conjectures using graphing calculator technology has been seamlessly incorporated into examples and exercises where appropriate. We recognize that the graphing calculator is only a tool to aid in the understanding of mathematics. Nowhere does this technology overshadow the mathematics; it simply augments the mathematics and allows the power and relevance of calculus to shine through.

Content Highlights

Applications

The applications in Brief Calculus: The Study of Rates of Change are the heart of our textbook. Reviewers have lauded the quantity, quality, and variety of applications in our text calling them superior to that of competitors. Application examples and exercises usually end with the phrase "...and interpret" so that students are asked not only to determine the numerical solution to an application, but also to write the meaning of the solution in the context of the application itself.

Many application models such as U.S. Imports from China, the number of cases of tuberculosis in the U.S., and the percentage of 3 to 5 year olds enrolled in preschool are derived from real data gathered from sources such as the U.S. Census Bureau web site and are indicated by the "On the Web" icon. We incorporate real life applications taken from business and economics, the biological and life sciences, and the social and physical sciences throughout the entire text.

Rate of Change Theme

In Chapter 1, the basic algebraic and transcendental functions are reviewed in a concise, yet comprehensive manner. These functions are used to introduce the core concept of Chapter 1, that the average rate of change of a function over an interval is equivalent to the slope of a secant line over an interval. Appropriate units and proper interpretation of solutions are stressed. For example, a solution to an application mad read, "This means during the period from 1987 to 1991, U.S. imports from China increased at a rate of 3.18 billion dollars per year." By introducing this idea 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 emphasize the rate of change theme, along with the importance of correct units and a reasonable interpretation of the solution. If any student who uses Brief Calculus: The Study of Rates of Change is asked, "What is calculus?", the student will inevitably proclaim "It is the study of rates of change!"

To supplement the rate of change theme, the differential is introduced in Chapter 3 and is used as a mathematical tool to introduce new topics in later sections. After a simple introduction and explanation, the differential is used to introduce linear approximations, marginal analysis, and for measuring rates and errors. In Chapter 4, the differential is used to derive the elasticity of demand formula.

Rule of Three (plus one)

Concepts such as functions, rates of change, limits, derivatives, marginal analysis, optimization, integrals, differential equations, and partial derivatives are analyzed numerically, graphically, and algebraically. In addition, we stress the verbal approach through an emphasis on interpretation of solutions. The Rule of Three (plus one) is particularly powerful in Chapter 2 when limits and the derivative are introduced.

Graphs and Art

Our experience in the classroom indicates that many students at the brief calculus level are visual learners. Because of this observation, our text contains an abundance of graphs, pictures, chart, and tables. This visual approach acts to reinforce the mathematical concepts of calculus and to show that many real life applications begin with numerical data. Moreover, the charts and tables visually demonstrate to students that they are working frequently with real data taken from genuine sources.

Exercises

For many instructors, the exercise sets in a textbook are one of the most important components. We have taken great care to provide exercises that meet user demand in terms of quantity and quality. With greater than 3500 exercises from which to choose, we believe instructors and students will have no problem finding the types of problems they need ranging from routine skill and practice type problems to multi-step skill and concept based applications. We use a combination of real data-based problems, with their inherent complexities, and realistic problems that offer beneficial, but tidier solutions.

The exercises have been carefully written in matching odd-even pairs and are graded by level of difficulty. Care was given to write the exercises and solutions in a style and approach that is consistent with the text. Each exercise has been scrutinized in every draft of the manuscript to insure accuracy and appropriateness.

Chapter Features

Chapter Openers

The first page of each chapter includes a photo and a pair of graphs that foreshadow the fundamental ideas to be presented in the chapter. "What We Know" reiterates what information has been learned in previous chapters and "Where Do We Go" tells what new concepts will be studied. Taken together, these features help create a road map to guide students through the book and underscore the connections between topics.

Flashbacks

Selected functions and examples used earlier in the textbook are revisited to introduce new concepts and are denoted by the Flashback icon. The Flashback concisely reviews an example from a previous section and then extends the problem by considering other questions that may be asked. In this manner, new topics are motivated in a natural way. Moreover, the Flashback often reviews skills and concepts from previous chapters that are needed. We believe that his pedagogical technique of using functions and applications previously introduced allows students to concentrate on new concepts using familiar applications.

On the Web

Many of the rich and varied applications in the textbook have been researched on the Internet. The "On the Web" icon, denotes applications which use models based on data gathered from the Internet. This feature impresses on students the fact that calculus can be applied to real world problems. We have performed extensive research to insure that we have included applications from a variety of disciplines including problems taken from business and economics, the social sciences, the biological and life sciences, and the physical sciences.

Interactive Activities

Extensions to worked out examples are denoted by the "Interactive Activity" icon. Many Interactive Activities are used to examine the problem from another perspective using the Rule of Three, while others explore additional properties of recently introduced topics. Others ask the student to do an exploration and make a conjecture to a completed example. Interactive Activities may serve many purposes: instructors may assign them as critical thinking exercises, they may be used as a springboard for classroom discussion, or may provide a vehicle for a collaborative activity. Solutions to selected Interactive Activities are given on the textbook's companion web site ( www.prenhall.com/armstrong ).

Checkpoints

At strategic points in each section, examples are followed by an exercise denoted with the "Checkpoint" icon. Each checkpoint asks students to work a particular, odd-numbered problem in the exercise set and helps to insure that a recently introduced skill or concept is understood. We have carefully chosen a parallel problem that requires a similar solution process to encourage students to check their grasp of the concept or skill. This pedagogical tool promotes better interaction between the text and student and encourages students to develop good study habits. Students who make use of the checkpoints will learn to take ownership of the course material.

Technology Notes

Additional tips and instructions for using graphing calculators are indicated with the "Technology Notes" icon. These notes do not give keystroke commands, instead they offer tips based on common questions that students may have. Some Technology Notes refer to the online calculator manuals found at the textbook's companion web site ( www.prenhall.com/armstrong ).

From Your Toolbox

Whenever a previously introduced key definition, theorem, or property is needed, it is quickly reviewed and denoted by the "From Your Toolbox" icon. This feature allows students to stay on task with the topic at hand without having to interrupt their reading to flip back to review previous material.

Notes

Immediately following many definitions, theorems, or properties, brief "Notes" are included to clarify a mathematical idea verbally, and to provide additional insights to help students understand the material.

Section Projects

At the end of each section, a Section Project presents a series of questions that ask students to explore the idea presented. We designed these questions to challenge, rather than discourage, the student. Many of the projects provide real data collected from the Internet and ask students to use the regression capabilities of their calculator to produce a model for the data, then to apply the recently introduced calculus concepts to the model. Instructors may use these section projects as a standard hand-in assignment or for a collaborative activity.

Supplements

Student's Solution Manual (0-13-085882-X)

Written by Matthew Hudock, Saint Philips College, San Antonio TX. This booklet contains complete, worked out solutions to all of the odd numbered exercises and review problems in the text.

Companion Web Site

This site ( www.prenhall.com/armstrong ) is designed to complement the text by offering a variety of teaching and learning resources including: a list of chapter -objectives, a readiness quiz for each chapter to help students assess their preparedness for the chapter contents, solutions to many of the Interactive Activities, a set of destinations with finks to other course related sites; the online graphing calculator manuals referenced in the Technology Notes, and a bulletin board for submitting and answering questions.

Instructor's Solution Manual (0-13-085885-4)

Written by Matthew Hudock, Saint Philips College, San Antonio TX. This booklet contains complete, worked out solutions to all even-numbered exercises and review problems in the text.

Test Item File (0-13-085881-1)

Written by Laurel Technical Service, Inc. This volume contains hardcopy of the test items available in PH Custom Test.

PH Custom Test: Windows (0-13-040295-8) Macintosh (0-13-040297-4) PH Custom Test is a menu-driven random test generator available on either a Windows or Macintosh platform. The system incorporates a unique editing function that allows the instructor to enter additional problems, or alter existing problems in the test bank using a full set of mathematical notation. The test system offers free-response, multiple-choice, and mixed exams. An almost unlimited number of quizzes, review exercises, and chapter tests may be generated quickly and easily. The system will also save time by producing answer keys, student worksheets, and a gradebook for the instructor, if desired

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