Brief Calculus: The Study of Rates of Change

Brief Calculus: The Study of Rates of Change, 1/e

William A. Armstrong
Donald E. Davis, both of Lakeland Community College

Coming December, 1999 by Prentice Hall Engineering/Science/Mathematics

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Applied Calculus-Mathematics

Summary

Appropriate for a 1 or 2 term course in Calculus for students majoring in Business, Economics, Social Science or Life Sciences. This is the first post-reform calculus text 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 the 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 text the ideal choice for math anxious students.

Features

Written for students.

Recognizing that many students in this course are uncomfortable with mathematics, the authors use a familiar, conversational writing style that mirrors most classroom presentations. Through their detailed explanations of topics and numerous study aides, the authors encourage students to interpret and understand concepts and not just “crunch numbers”.

Outstanding Applications in examples and exercise sets. Reviewers have lauded the quality, quantity, and variety of applications in the text calling them “superior” to competitors.

Many real data based applications (U.S. Imports from China, population data models, pharmacological studies on medicine in the blood stream, etc.) are interspersed with simpler, but realistic, applications to demonstrate how the mathematical concepts are applied in various situations. Section 7.4 alone includes over 20 applications of the notoriously dry content of integration by parts.

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.

This idea is central to the study of differential calculus. Most students lack an intuitive understanding of “rate of change” and consequently struggle with the more abstract ideas of calculus. Thus, by first introducing rate of change in terms of slope, a concept with which students are familiar, they are less intimidated when they study the derivative.

Introduction to Limits (Section 2.1).

Armstrong/Davis ease the students into the idea of limits at the beginning of chapter 2 with a well paced discussion that introduces left and right hand limits to motivate the limit concept. The authors use the combined numerical, graphical, and algebraic (“rule of three”) approach that allows them to introduce the notion of limits intuitively before presenting a more formal definition.
Professors agree that a thorough understanding of the limit concept is one of the most important ideas students should master in this course.

Section 2.2 introduces limits at infinity and infinite limits so that a variety of interesting, real life applications of the limit concept may be provided early in this chapter. These applications include the cost of removing pollutants from city's lake, the growth of a flu epidemic in Spring Point, the amount of medication in a patient's body, etc.

The varied applications demonstrate the relevance of the topic to students and show how they are likely to use the mathematics in future employment.

A distinctive Flashback feature appears throughout the text to recall previous exercises and examples and to underscore the links between old and new material.

Reviewers agree this feature is a major strength of the text that benefits the student in many ways. By reinforcing ideas previously learned and repeatedly demonstrating the importance of those topics, students attain a certain comfort level with the problem after visiting it a second or third time.

Section 2.3 begins with a Flashback (U.S. imports from China) application that uses rate of change to develop the notion of the derivative. The U.S. imports application, along with others, is woven through the chapter to extend the rate of change theme.

This carefully paced section couched in the framework of a familiar application and idea (rate of change) makes the introduction to the derivative concept easier to understand and retain.

In contrast to many competitors, Armstrong/Davis include a strong introduction to the differential in section 3.1. Section 3.2 follows with marginal business functions which are a natural by-product of the differential.

These sections demonstrate the relationship between these essential business functions and the underlying mathematics. Courses with numerous business majors will applaud this coverage.

This text devotes two sections (5.4 and 5.5) to the difficult concept of optimization.

By slowing down the pace of the presentation and incorporating numerous applications, students gain a better understanding of this challenging material.

From the Toolbox is another reinforcement feature which is unique to this text.

Appearing as needed, the information in these boxes reviews and reinforces important concepts which students tent to forget but which are essential to understanding the concept under consideration.

Seamless integration of the Graphing Calculator.

Output screens from the TI-83 Graphing Calculator are integrated throughout the text and explanations often reference the use of a graphing utility. Marginal Technology Notes provide additional insights and refer students to the companion website for keystroke level instruction.

Check Points, integrated throughout the text, direct students to work target exercises in the section problem sets that correspond directly to the idea that has been presented.

Working the Checkpoint exercises as directed fosters good study skills and provides the immediate reinforcement and practice necessary to master the mathematical concepts and techniques.

Outstanding variety and choice in the Exercise Sets.

Greater than 3000 problems offer a wealth of material from which to choose. Exercises range from simple drill and skill type problems to check mastery of basic techniques to multi-step real data based applications that test the student's conceptual understanding. Each section ends with a Section Project that extends knowledge in the text to a real world application. Section Projects may effectively be assigned as a group project.

Many applications, examples, and exercises are accompanied by an “On the Web” icon to denote that the supporting raw data is available from a web based source.

The companion website includes links to these sites for students or professors who wish to delve more deeply into the subject.

Another feature that encourages student exploration is the Interactive Activity.

Integrated throughout the text, the Interactive Activities encourage students to extend the material presented. Many of these exercises serve as de facto graphing calculator problems. Solutions to the Interactive Activities are posted on the companion website.