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More About This Textbook
Overview
This introduction to mathematical modeling uses elementary functions to describe and explore realworld data and phenomena. Helps readers connect math with the world around them through realworld applications of elementary mathematics. Shows how to construct useful mathematical models, how to analyze them critically, and how to communicate quantitative concepts effectively. Uses concrete language and examples throughout to foster quantitative literacy. For anyone interested in gaining a solid foundation in mathematical concepts.
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Meet the Author
MARY ELLEN DAVIS Georgia Perimeter College, received her Master of Arts degree in mathematics from the University of MissouriColumbia in 1976. She has taught mathematics at the secondary level and at Georgia State University and the University of Birmingham (England). She joined the mathematics department at Georgia Perimeter College (then DeKalb College) in 1991 and has taught a wide range of courses from college algebra to calculus and statistics. She was instrumental in the piloting and implementing of the college's Introduction to Mathematical Modeling course in 1998. She was selected as a Georgia Governor's Teaching Fellow in 1996, and in 1999 received a GPC Distance Education Fellowship to develop webbased materials for applied calculus.
C. HENRY EDWARDS (Ph.D. University of Tennessee) Emeritus professor of mathematics at the University of Georgia, Edwards recently retired after 40 years of undergraduate classroom teaching at the universities of Tennessee, Wisconsin, and Georgia. Although respected for his diverse research interests, Edwards' first love has always remained teaching. Throughout his teaching career he has received numerous college and universitywide teaching awards, including the University of Georgia's honoratus medal in 1983 and its Josiah Meigs award in 1991. In 1997, Edwards was the first universitylevel faculty recipient of the Georgia Board of Regents newlyinstituted statewide award for teaching excellence.
A prolific author, Edwards is coauthor of wellknown calculus and differential equations textbooks and has written a book on the history of mathematics, in addition to several instructionalcomputer manuals. During the 1990s, Edwards has worked on three NSFsupported projects that fostered a better integration of technology into the mathematics curriculum. The last three years of his long teaching career were devoted principally to the development of a new technologyintensive entrylevel mathematics course on which this new textbook is based. Additional information is provided on his web page www.math.uga.edu/~hedwards.
Read an Excerpt
PREFACE
This textbook is for an entrylevel college mathematics course at the same academic level as college algebra, but intended for students who are not necessarily preparing for subsequent courses in calculus. Our approach is based on the exploitation of graphingcalculator technology to engage students in concrete modeling applications of mathematics. The mathematical ideas of the course center on functions and their graphs—ranging from linear functions and polynomials to exponential and trigonometric functions—that we hope will seem familiar and friendly to students who complete the course.
BRIEF DESCRIPTION
Specifically, this textbook presents an introduction to mathematical modeling based on the use of elementary functions to describe and explore realworld data and phenomena. It demonstrates graphical, numerical, symbolic, and verbal approaches to the investigation of data, functions, equations, and models. We emphasize interesting applications of elementary mathematics together with the ability to construct useful mathematical models, to analyze them critically, and to communicate quantitative concepts effectively. In short, this is a textbook for
RATIONALE FOR A NEW COURSE
The content of the traditional college algebra course is defined largely by the paperandpencil skills (mainly symbolic manipulation) that are needed by students whose curricula point them towards a subsequent calculus course.However, many of the students in a typical college algebra course are not really headed for calculus or never make it there. For too many of these students, college algebra consists of revisiting the skills and concepts, either mastered or not, which were "covered" in several previous mathematics courses. This experience leaves students with little enhancement of the quantitative skills they most need for their subsequent studies. It is a missed opportunity for them to begin college with a useful mathematics course that is interesting both to students and to instructors, and which offers a solid chance for progress and success.
There is wide agreement on the need for an alternative new approach to fill this void. Both the NCTM's Principles and Standards for School Mathematics and AMATYC's Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus recommend that mathematics courses teach students to reason mathematically, to model realworld situations, and to make use of appropriate technologies. We offer this as an appropriate textbook for such a course. The evolution of these materials began with a web site that was originally developed (starting in 1996) to support University of Georgia students taking pilot sections of this new course. About two thousand students have now used preliminary versions of the textbook. Many of these students have reacted with enthusiasm belying their typical lack of success in prior mathematical experiences. We hope this apparent success and satisfaction will carry over to the students who use this published textbook.
PURPOSE AND OBJECTIVES
The primary objective of this new course is the development of the quantitative literacy and savvy that college graduates need to function effectively in society and workplace. The course exploits technology and realworld applications to motivate necessary skill development and the ability to reason and communicate mathematically, to use elementary mathematics to solve applied problems, and to make connections between mathematics and the surrounding world.
With a flavor combining functions and graphs with data modeling, the course is based largely on the use of graphing calculator methods in lieu of traditional symbolic manipulations to solve both familiar and nonstandard problems. The focus of the course is "mathematical modeling" and the use of elementary mathematics—numbers and measurement, algebra, geometry, and data exploration—to investigate realworld problems and questions.
As an alternative to the standard college algebra course—though at the same academic level—this course is intended for students who are not necessarily headed for calculusbased curricula, but still need a solid quantitative foundation both for subsequent studies and for life as educated citizens and workers. Graphing technology enables these students to experience the power of mathematics and to enjoy success in solving interesting and significant problems (an experience that they all too rarely enjoy in traditional college algebra courses).
PURPOSE AND OBJECTIVES
The book consists of the following chapters:
Table of Contents
(NOTE: Each chapter begins with a realworld vignette that is revisited as the chapter evolves, and concludes with a Review and a projectstyle Investigation.)
1. Functions and Mathematical Models.
Functions Defined by Tables. Functions Defined by Graphs. Functions Defined by Formulas. Average Rate of Change.
2. Linear Functions and Models.
Constant Change and Linear Growth, Linear Functions and Graphs. PiecewiseLinear Functions. Fitting Linear Models to Data.
3. Natural Growth Models.
Percentage Growth and Interest. Percentage Decrease and HalfLife, Natural Growth and Decline in the World. Fitting Natural Growth Models to Data.
4. Exponential and Logarithmic Models.
Compound Interest and Continuous Growth. Exponential and Logarithmic Functions. Exponential and Logarithmic Data Modeling.
5. Quadratic Functions and Models.
Quadratic Functions and Graphs. Quadratic Highs and Lows. Fitting Quadratic Models to Data.
6. Polynomial Models and Linear Systems.
Solving Polynomial Equations. Solving Pairs of Linear Equations—Lots of Ways! Linear Systems of Equations. Polynomial Data Modeling.
7. Bounded Growth Models.
Limited Populations. Fitting Logistic Models to Data. Discrete Models and Chaos.
8. Trigonometric Models.
Periodic Phenomena and Trigonometric Functions. Trigonometric Data Modeling.
Answers to Selected Problems.
Index.
Credits.
Preface
PREFACE
This textbook is for an entrylevel college mathematics course at the same academic level as college algebra, but intended for students who are not necessarily preparing for subsequent courses in calculus. Our approach is based on the exploitation of graphingcalculator technology to engage students in concrete modeling applications of mathematics. The mathematical ideas of the course center on functions and their graphs—ranging from linear functions and polynomials to exponential and trigonometric functions—that we hope will seem familiar and friendly to students who complete the course.
BRIEF DESCRIPTION
Specifically, this textbook presents an introduction to mathematical modeling based on the use of elementary functions to describe and explore realworld data and phenomena. It demonstrates graphical, numerical, symbolic, and verbal approaches to the investigation of data, functions, equations, and models. We emphasize interesting applications of elementary mathematics together with the ability to construct useful mathematical models, to analyze them critically, and to communicate quantitative concepts effectively. In short, this is a textbook for
RATIONALE FOR A NEW COURSE
The content of the traditional college algebra course is defined largely by the paperandpencil skills (mainly symbolic manipulation) that are needed by students whose curricula point them towards a subsequent calculuscourse.However, many of the students in a typical college algebra course are not really headed for calculus or never make it there. For too many of these students, college algebra consists of revisiting the skills and concepts, either mastered or not, which were "covered" in several previous mathematics courses. This experience leaves students with little enhancement of the quantitative skills they most need for their subsequent studies. It is a missed opportunity for them to begin college with a useful mathematics course that is interesting both to students and to instructors, and which offers a solid chance for progress and success.
There is wide agreement on the need for an alternative new approach to fill this void. Both the NCTM's Principles and Standards for School Mathematics and AMATYC's Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus recommend that mathematics courses teach students to reason mathematically, to model realworld situations, and to make use of appropriate technologies. We offer this as an appropriate textbook for such a course. The evolution of these materials began with a web site that was originally developed (starting in 1996) to support University of Georgia students taking pilot sections of this new course. About two thousand students have now used preliminary versions of the textbook. Many of these students have reacted with enthusiasm belying their typical lack of success in prior mathematical experiences. We hope this apparent success and satisfaction will carry over to the students who use this published textbook.
PURPOSE AND OBJECTIVES
The primary objective of this new course is the development of the quantitative literacy and savvy that college graduates need to function effectively in society and workplace. The course exploits technology and realworld applications to motivate necessary skill development and the ability to reason and communicate mathematically, to use elementary mathematics to solve applied problems, and to make connections between mathematics and the surrounding world.
With a flavor combining functions and graphs with data modeling, the course is based largely on the use of graphing calculator methods in lieu of traditional symbolic manipulations to solve both familiar and nonstandard problems. The focus of the course is "mathematical modeling" and the use of elementary mathematics—numbers and measurement, algebra, geometry, and data exploration—to investigate realworld problems and questions.
As an alternative to the standard college algebra course—though at the same academic level—this course is intended for students who are not necessarily headed for calculusbased curricula, but still need a solid quantitative foundation both for subsequent studies and for life as educated citizens and workers. Graphing technology enables these students to experience the power of mathematics and to enjoy success in solving interesting and significant problems (an experience that they all too rarely enjoy in traditional college algebra courses).
PURPOSE AND OBJECTIVES
The book consists of the following chapters: