Brief Calculus with Applications

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9780130655912: Brief Calculus with Applications

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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About 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, along with 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.

Excerpt. © Reprinted by permission. All rights reserved.:


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 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 2.3Problem 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 6.4Problem 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.


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, 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,
  • The annual total expenditures on pollution abatement in the U.S. (Source: U.S. Statistical Abstract,
  • The rate of change in the number of arrests for drug abuse violations by American adults (Source: U.S. Census Bureau,

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

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.


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Armstrong, Bill; Davis, Don
Published by Prentice Hall (2002)
ISBN 10: 0130655910 ISBN 13: 9780130655912
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