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C. Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960, and recently retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin, and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia’s honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution’s highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s, he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with-Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students.
David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 (under the direction of Prof. L. Bruce Treybig) while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran’s Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee’s research team’s primary focus was on the active transport of sodium ions by biological membranes. Penney’s primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms. Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia, he received numerous University-wide teaching awards as well as directing several doctoral dissertations and seven undergraduate research projects. He is the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra, and liberal arts mathematics.
About the Authors | |
Preface | |
Functions, Graphs, and Models | |
Functions and Mathematical ModelingInvestigation: Designing a Wading Pool | |
Graphs of Equations and Functions | |
Polynomials and Algebraic Functions | |
Transcendental Functions | |
Preview: What Is Calculus? | |
Review - Understanding: Concepts and Definitions | |
Objectives: Methods and Techniques | |
Prelude to Calculus | |
Tangent Lines and Slope Predictors | |
Investigation: Numerical Slope Investigations | |
The Limit ConceptInvestigation: Limits, Slopes, and Logarithms | |
More About LimitsInvestigation: Numerical Epsilon-Delta Limits | |
The Concept of Continuity | |
Review - Understanding: Concepts and Definitions | |
Objectives: Methods and Techniques | |
The Derivative | |
The Derivative and Rates of Change | |
Basic Differentiation Rules | |
The Chain Rule | |
Derivatives of Algebraic Functions | |
Maxima and Minima of Functions on Closed Intervals | |
Investigation: When Is Your Coffee Cup Stablest? | |
Applied Optimization Problems | |
Derivatives of Trigonometric Functions | |
Exponential and Logarithmic Functions | |
Investigation: Discovering the Number e for Yourself | |
Implicit Differentiation and Related Rates | |
Investigation: Constructing the Folium of Descartes | |
Successive Approximations and Newton's Method | |
Investigation: How Deep Does a Floating Ball Sink? | |
Review - Understanding: Concepts, Definitions, and Formulas | |
Objectives: Methods and Techniques | |
Additional Applications of the Derivative | |
Introduction | |
Increments, Differentials, and Linear Approximation | |
Increasing and Decreasing Functions and the Mean Value Theorem | |
The First Derivative Test and Applications | |
Investigation: Constructing a Candy Box With Lid | |
Simple Curve Sketching | |
Higher Derivatives and Concavity | |
Curve Sketching and Asymptotes | |
Investigation: Locating Special Points on Exotic Graphs | |
Indeterminate Forms and L'Hapital's Rule | |
More Indeterminate Forms | |
Review - Understanding: Concepts, Definitions, and Results | |
Objectives: Methods and Techniques | |
The Integral | |
Introduction | |
Antiderivatives and Initial Value Problems | |
Elementary Area Computations | |
Riemann Sums and the Integral | |
Investigation: Calculator/Computer Riemann Sums | |
Evaluation of Integrals | |
The Fundamental Theorem of Calculus | |
Integration by Substitution | |
Areas of Plane Regions | |
Numerical IntegrationInvestigation: Trapezoidal and Simpson Approximations | |
Review - Understanding: Concepts, Definitions, and Results | |
Objectives: Methods and Techniques | |
Applications of the Integral | |
Riemann Sum Approximations | |
Volumes by the Method of Cross Sections | |
Volumes by the Method of Cylindrical Shells | |
Investigation: Design Your Own Ring! | |
Arc Length and Surface Area of Revolution | |
Force and Work | |
Centroids of Plane Regions and Curves | |
The Natural Logarithm as an Integral | |
Investigation: Natural Functional Equations | |
Inverse Trigonometric Functions | |
Hyperbolic Functions | |
Review - Understanding: Concepts, Definitions, and Formulas | |
Objectives: Methods and Techniques | |
Techniques of Integration | |
Introduction< | |
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