
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
Copyright 2000, 850 pp.
Cloth
ISBN 0137549040

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Applied CalculusMathematics

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 postreform 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.
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 byproduct
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 TI83 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 multistep 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.
1. Functions, Modeling and Average Rate of Change.
2. Limits, Instantaneous Rate of Change and the Derivative.
3. Applications of the Derivative.
4. Additional Differentiation Techniques.
5. Further Applications of the Derivative
6. Integral Calculus
7. Applications of Integral Calculus.
8. Calculus of Several Variables
Appendices.
A. Essentials of Algebra.
B. Multiple Regression Program.
C. Selected Proofs.
