Calculus, Early Transcendentals

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.

TABLE OF CONTENTS    

                                                                                   

About the Authors

Preface

 

1 Functions, Graphs, and Models

            1.1 Functions and Mathematical Modeling

            Investigation:   Designing a Wading Pool  

            1.2 Graphs of Equations and Functions

            1.3 Polynomials and Algebraic Functions

            1.4 Transcendental Functions

            1.5 Preview:  What Is Calculus?

            REVIEW — Understanding: Concepts and Definitions

            Objectives:  Methods and Techniques    

 

2 Prelude to Calculus

            2.1 Tangent Lines and Slope Predictors

            Investigation:   Numerical Slope Investigations  

            2.2 The Limit Concept

            Investigation:   Limits, Slopes, and Logarithms  

            2.3 More About Limits

            Investigation:   Numerical Epsilon-Delta Limits  

            2.4 The Concept of Continuity

            REVIEW – Understanding: Concepts and Definitions

            Objectives:  Methods and Techniques 

 

3 The Derivative

            3.1 The Derivative and Rates of Change

            3.2 Basic Differentiation Rules

            3.3 The Chain Rule 

            3.4 Derivatives of Algebraic Functions

            3.5 Maxima and Minima of Functions on Closed Intervals

            Investigation:   When Is Your Coffee Cup Stablest?  

            3.6 Applied Optimization Problems

            3.7 Derivatives of Trigonometric Functions

            3.8 Exponential and Logarithmic Functions 

            Investigation:   Discovering the Number  e  for Yourself 

            3.9 Implicit Differentiation and Related Rates

            Investigation:   Constructing the Folium of Descartes 

            3.10 Successive Approximations and Newton's Method

            Investigation:   How Deep Does a Floating Ball Sink? 

            REVIEW — Understanding: Concepts, Definitions, and Formulas

            Objectives:  Methods and Techniques  

 

4 Additional Applications of the Derivative

            4.1 Introduction

            4.2 Increments, Differentials, and Linear Approximation

            4.3 Increasing and Decreasing Functions and the Mean Value Theorem

            4.4 The First Derivative Test and Applications

            Investigation:   Constructing a Candy Box With Lid  

            4.5 Simple Curve Sketching

            4.6 Higher Derivatives and Concavity

            4.7 Curve Sketching and Asymptotes

            Investigation:   Locating Special Points on Exotic Graphs  

            4.8 Indeterminate Forms and L'Hôpital's Rule

            4.9 More Indeterminate Forms

            REVIEW – Understanding: Concepts, Definitions, and Results

            Objectives:  Methods and Techniques   

 

5 The Integral  

            5.1 Introduction

            5.2 Antiderivatives and Initial Value Problems

            5.3 Elementary Area Computations

            5.4 Riemann Sums and the Integral

            Investigation:   Calculator/Computer Riemann Sums  

            5.5 Evaluation of Integrals

            5.6 The Fundamental Theorem of Calculus

            5.7 Integration by Substitution

            5.8 Areas of Plane Regions

            5.9 Numerical Integration

            Investigation:   Trapezoidal and Simpson Approximations 

            REVIEW — Understanding: Concepts, Definitions, and Results

            Objectives:  Methods and Techniques   

 

6 Applications of the Integral

            6.1 Riemann Sum Approximations

            6.2 Volumes by the Method of Cross Sections

            6.3 Volumes by the Method of Cylindrical Shells

            Investigation:   Design Your Own Ring!

            6.4 Arc Length and Surface Area of Revolution

            6.5 Force and Work

            6.6 Centroids of Plane Regions and Curves

            6.7 The Natural Logarithm as an Integral

            Investigation:   Natural Functional Equations

            6.8 Inverse Trigonometric Functions

            6.9 Hyperbolic Functions

            REVIEW – Understanding: Concepts, Definitions, and Formulas

            Objectives:  Methods and Techniques        

 

7 Techniques of Integration 

            7.1 Introduction

            7.2 Integral Tables and Simple Substitutions

            7.3 Integration by Parts

            7.4 Trigonometric Integrals

            7.5 Rational Functions and Partial Fractions

            7.6 Trigonometric Substitutions

            7.7 Integrals Involving Quadratic Polynomials

            7.8 Improper Integrals

            SUMMARY — Integration Strategies 

            REVIEW – Understanding: Concepts and Techniques

            Objectives:  Methods and Techniques    

 

8 Differential Equations    

            8.1 Simple Equations and Models

            8.2 Slope Fields and Euler's Method

            Investigation:   Computer-Assisted Slope Fields and Euler's Method 

            8.3 Separable Equations and Applications

            8.4 Linear Equations and Applications

            8.5 Population Models

            Investigation:   Predator-Prey Equations and Your Own Game Preserve

            8.6 Linear Second-Order Equations

            8.7 Mechanical Vibrations

            REVIEW — Understanding: Concepts, Definitions, and Methods

            Objectives:  Methods and Techniques    

 

9 Polar Coordinates and Parametric Curves  

            9.1 Analytic Geometry and the Conic Sections

            9.2 Polar Coordinates

            9.3 Area Computations in Polar Coordinates

            9.4 Parametric Curves

            Investigation:   Trochoids Galore

            9.5 Integral Computations with Parametric Curves

            Investigation:   Moon Orbits and Race Tracks

            9.6 Conic Sections and Applications 

            REVIEW – Understanding: Concepts, Definitions, and Formulas

            Objectives:  Methods and Techniques 

 

10 Infinite Series  

            10.1 Introduction

            10.2 Infinite Sequences

            Investigation:   Nested Radicals and Continued Fractions

            10.3 Infinite Series and Convergence

            Investigation:   Numerical Summation and Geometric Series

            10.4 Taylor Series and Taylor Polynomials

            Investigation:   Calculating Logarithms on a Deserted Island 

            10.5 The Integral Test

            Investigation:   The Number  p, Once and for All

            10.6 Comparison Tests for Positive-Term Series

            10.7 Alternating Series and Absolute Convergence

            10.8 Power Series

            10.9 Power Series Computations

            Investigation:   Calculating Trigonometric Functions on a Deserted Island  

            10.10 Series Solutions of differential Equations

            REVIEW — Understanding: Concepts, and Results

            Objectives:  Methods and Techniques    

 

11 Vectors, Curves, and Surfaces in Space

            11.1 Vectors in the Plane

            11.2 Three-Dimensional Vectors

            11.3 The Cross Product of Two Vectors

            11.4 Lines and Planes in Space

            11.5 Curves and Motion in Space

            Investigation:   Does a Pitched Baseball Really Curve?

            11.6 Curvature and Acceleration

            11.7 Cylinders and Quadric Surfaces

            11.8 Cylindrical and Spherical Coordinates

            REVIEW – Understanding: Concepts, Definitions, and Results

            Objectives:  Methods and Techniques    





                                    



12 Partial Differentiation

            12.1 Introduction

            12.2 Functions of Several Variables

            12.3 Limits and Continuity

            12.4 Partial Derivatives

            12.5 Multivariable Optimization Problems

            12.6 Increments and Linear Approximation

            12.7 The Multivariable Chain Rule

            12.8 Directional Derivatives and the Gradient Vector

            12.9 Lagrange Multipliers and Constrained Optimization

            Investigation:   Numerical Solution of Lagrange Multiplier Systems

            12.10 Critical Points of Functions of Two Variables

            Investigation:   Critical Point Investigations

            REVIEW — Understanding: Concepts, Definitions, and Results

            Objectives:  Methods and Techniques    

 

13 Multiple Integrals      

            13.1 Double Integrals

            Investigation:   Midpoint Sums Approximating Double Integrals 

            13.2 Double Integrals over More General Regions

            13.3 Area and Volume by Double Integration

            13.4 Double Integrals in Polar Coordinates

            13.5 Applications of Double Integrals

            Investigation:   Optimal Design of Race Car Wheels 

            13.6 Triple Integrals

            Investigation:   Archimedes' Floating Paraboloid

            13.7 Integration in Cylindrical and Spherical Coordinates

            13.8 Surface Area

            13.9 Change of Variables in Multiple Integrals

            REVIEW – Understanding: Concepts, Definitions, and Results

            Objectives:  Methods and Techniques    

 

14 Vector Calculus  

            14.1 Vector Fields

            14.2 Line Integrals

            14.3 The Fundamental Theorem and Independence of Path

            14.4 Green's Theorem

            14.5 Surface Integrals

            Investigation:   Surface Integrals and Rocket Nose Cones 

            14.6 The Divergence Theorem

            14.7 Stokes' Theorem

            REVIEW — Understanding: Concepts, Definitions, and Results

            Objectives:  Methods and Techniques    

 

Appendices

A:         Real Numbers and Inequalities

B:         The Coordinate Plane and Straight Lines

C:         Review of Trigonometry

D:         Proofs of the Limit Laws

E:         The Completeness of the Real Number System

F:         Existence of the Integral

G:         Approximations and Riemann Sums

H:         L'Hôpital's Rule and Cauchy's Mean Value Theorem

I:          Proof of Taylor's Formula

J:          Conic Sections as Sections of a Cone

K:        Proof of the Linear Approximation Theorem

L:         Units of Measurement and Conversion Factors

M:        Formulas from Algebra, Geometry, and Trigonometry

N:        The Greek Alphabet

 

Answers to Odd-Numbered Problems

References for Further Study

Index