Features What's New Ancillaries Table of Contents

Calculus, Single and Multivariable, Second Edition

Deborah Hughes-Hallett and Andrew M. Gleason, both of Harvard University
William G. McCallum of University of Arizona, et al.
ISBN: 0-471-19490-5, 984 pages, cloth, Combined Version, 1998.

A revision of the best selling innovative Calculus text on the market. A product of an NSF funded grant, this next generation text is concise in number of topics and lenth. Functions are presented graphically, numerically, algebraically, and verbally to give students the benefit of alternate interpretations. The text is problem driven with exceptional exercises based on real world applications from engineering, physics, life sciences, and economics. Technology is used as a tool to help students learn to think mathematically. The second edition continues its focused vision by covering only the essential calculus topics students need for later courses.

Features

• New thread of honors sections, called Focus on Theory, that build upon the student's geometric intuition. These sections present case studies in the formulation of axioms, definitions, and proof, including the completeness axiom, the definition of limit, the Extreme Value Theorem, and the Mean Value Theorem.

• New section on limits and continuity in Chapter 1.

• New section on limits, l'Hopital's Rule, and relative growth rates in Chapter 4.

• New section on hyperbolic functions in Chapter 5.

• Reorganization of some material: properties of the definite integral have been moved to Chapter 3, refocusing Chapter 6 on reconstructing functions from their derivatives.

• Chapter 9 (formerly Chapter 10) on Approximations and Series has been expanded to include more discussion of series of constants. New section on ratio test and the interval of convergence of a power series, including a discussion of the harmonic series and alternating series. Differential equations is now Chapter 10.

• Inclusion of more easy and medium level problems in each section of the text, and a battery of practice problems at the end of Chapter 4 (differentiation) and Chapter 7 (integration).

• Answers are included in the back of the book to those odd numbered problems that have them.

• Each chapter includes a summary of the main points.

• Projects are included at the end of each chapter, as well as a wider selection in the appendix.

• Multivariable calculus may be taught with or without the vector calculus in Chapters 18, 19, and 20.

• Chapter 11. Section 11.7 on Limits and Continuity has been moved to a Focus on Theory section.

• Chapter 13. Section 13.8 on Partial Differential Equations has been omitted; Section 13.10 on Differentiability has been moved to a Focus on Theory section.

• Chapter 14. The material in Section 14.3 on the Lagrangian and on the interpretation of ( has been moved to a Focus on Modeling section.

• Chapter 15. Section 15.4 on the Monte Carlo method has been omitted. Section 15.8, Notes on Change of Variables, has been moved to a Focus on Theory section.

• Chapter 16. This chapter now focuses on Parameterized Curves. Sections 16.3 and 16.4 on Parameterized Surfaces and the Implicit Function Theorem have been omitted. Some of the material on Kepler's Laws has been moved to a project.

• Chapter 18. The proof of Green's Theorem in Section 18.5 has been moved to a Focus on Theory section.

• Chapter 19. Section 19.3 on Flux integrals over parameterized surfaces has been omitted.

• Chapter 20. Section 20.5 on the Three Fundamental Theorems has been moved to a Focus on Theory section; Section 20.6 containing the proofs of the Divergence and Stokes' Theorems has been omitted.

What's New

A problem driven text in which the student starts with a practical problem and derives general results from it.

The topical coverage was determined after discussion with mathematicians and client disciplines to focus on students'calculus needs. Emphasis is on depth of understanding and not breadth of coverage.

Equal weight is given to describing functions graphically, numerically, symbollically, and verbally in order to provide alternate avenues through which students can understand the material.

Many of the real world problems are open-ended, meaning that there may be more than one approach and more than one solution, depending on the student's analysis. Solving a problem usually relies on the use of common sense and critical thinking skills. Students are encouraged to develop estimating and approximating skills.

The book contains the main ideas of calculus in plain English to improve the students' understanding and to encourage them to read it.

Worked out examples are not entirely like the problem sets in the text. The result is that students cannot simply mimic techniques from the examples, without actually learning the concepts.

Technology is used as a tool to help students visualize the concepts. It is recommended that a graphing calculator, graphing software, or computer algebra system be used with this book but the emphasis is on the calculus concepts and not the technology.

Ancillaries

Instructor's Solutions Manual for Calculus: Single Variable, Second Edition (ISBN: 0-471-23910-0) Instructor's Solutions Manual for Calculus: Single and Multivariable, Second Edition (ISBN: 0-471-24294-2) Contains complete solutions to all problems.

Instructor's Manual for Calculus: Single and Multivariable, Second Edition (ISBN: 0-471-23909-7) Includes teaching tips by section, test questions by section, calculator programs, additional projects by section, and overhead transparency masters. Also available online.

Student Solutions Manual for Calculus: Single Variable, Second Edition (ISBN: 0-471-24294-2). Student Solutions Manual for Calculus: Single and Multivariable, Second Edition (ISBN: 0-471-24293-4). Contains complete solutions to all odd-numbered problems.

Instructors' Resource CD-ROM (ISBN: 0-471-24292-6) Contains all print supplements for both Calculus: Single Variable and Calculus: Single and Multivariable on CD-ROM.

Wiley Web Tests for Calculus (ISBN: 0-471-29357-1) Developed by John Orr and the University of Nebraska, Lincoln, the Web Tests are an Online testing system for use in Calculus I and II. This customizable and expandable testing system allows for administering and grading tests in a variety of modes. The software is platform independent and resides on the local server.

Calculus Live for Calculus: Single Variable, Second Edition (ISBN: 0-471-29381-4) Calculus Live for Calculus: Single and Multivariable, Second Edition (ISBN: 0-471-31788-8) Highly interactive full-text version of Hughes-Hallett driven by a Mathematica engine on CD-ROM. Provides a user-friendly front end that requires no knowledge of Mathematica syntax for imputing functions.

Table of Contents

1.  A LIBRARY OF FUNCTIONS
1.1   WHAT'S A FUNCTION?
1.2   LINEAR FUNCTIONS
1.3   EXPONENTIAL FUNCTIONS
1.4   POWER FUNCTIONS
1.5   NOTES ON INVERSE FUNCTIONS
1.6   LOGARITHMS
1.7   THE NUMBER e AND NATURAL LOGARITHMS
1.8   NEW FUNCTIONS FROM OLD
1.9   THE TRIGONOMETRIC FUNCTIONS
1.10 POLYMONIALS AND RATIONAL FUNCTIONS
1.11 INTRODUCTION TO CONTINUITY
REVIEW PROBLEMS
PROJECTS

FOCUS ON THEORY
UNDERPINNINGS OF CALCULUS
THE BINOMIAL THEOREM
COMPLETENESS OF THE REAL NUMBERS

2.  KEY CONCEPT:  THE DERIVATIVE
2.1   HOW DO WE MEASURE SPEED?
2.2   THE DERIVATIVE AT A POINT
2.3   THE DERIVATIVE FUNCTION
2.4   INTERPRETATIONS OF THE DERIVATIVE
2.5   THE SECOND DERIVATIVE
REVIEW PROBLEMS
PROJECTS

FOCUS ON THEORY
LIMITS AND CONTINUITY
DIFFERENTIABILITY AND LINEAR APPROXIMATION

3.  KEY CONCEPT:  THE DEFINITE INTEGRAL
3.1   HOW DO WE MEASURE DISTANCE TRAVELED?
3.2   THE DEFINITE INTEGRAL
3.3   INTERPRETATIONS OF THE DEFINITE INTEGRAL
3.4   THEOREMS ABOUT DEFINITE INTEGRALS
REVIEW PROBLEMS
PROJECTS

FOCUS ON THEORY
THE DEFINITE INTEGRAL

4.  SHORT-CUTS TO DIFFERENTIATION
4.1   POWERS AND POLYNOMIALS
4.2   THE EXPONENTIAL FUNCTION
4.3   THE PRODUCT AND QUOTIENT RULES
4.4   THE CHAIN RULE
4.5   THE TRIGONOMETRIC FUNCTIONS
4.6   APPLICATIONS OF THE CHAIN RULE
4.7   IMPLICIT FUNCTIONS
4.8   LINEAR APPROXIMATIONS AND LIMITS
REVIEW PROBLEMS
PROJECTS

FOCUS ON PRACTICE
DIFFERENTIATION

5.  USING THE DERIVATIVE
5.1   USING FIRST AND SECOND DERIVATIVES
5.2   FAMILIES OF CURVES:  A QUALITATIVE STUDY
5.3   OPTIMIZATION
5.4   APPLICATIONS TO MARGINALITY
5.5   MORE OPTIMIZATION:  INTRODUCTION TO MODELING
5.6   HYPERBOLIC FUNCTIONS
REVIEW PROBLEMS
PROJECTS

FOCUS ON THEORY
THEOREMS ABOUT CONTINUOUS AND DIFFERENTIABLE FUNCTIONS

6.  CONSTRUCTING ANTIDERIVATIVES
6.1   ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY
6.2   CONSTRUCTING ANTIDERIVATIVES ANALYTICALLY
6.3   DIFFERENTIAL EQUATIONS
6.4   SECOND FUNDAMENTAL THEOREM OF CALCULUS
REVIEW PROBLEMS
PROJECTS

FOCUS ON MODELING
THE EQUATIONS OF MOTION

7.  INTEGRATION
7.1   INTEGRATION BY SUBSTITUTION PART I
7.2   INTEGRATION BY SUBSTITUTION: PART II
7.3   INTEGRATION BY PARTS
7.4   TABLES OF INTEGRALS
7.5   APPROXIMATING DEFINITE INTEGRALS
7.6   APPROXIMATION ERRORS AND SIMPSON'S RULE
7.7   IMPROPER INTEGRALS
7.8   MORE IMPROPER INTEGRALS
REVIEW PROBLEMS
PROJECTS

FOCUS ON PRACTICE
INTEGRATION

8.  USING THE DEFINITE INTEGRAL
8.1   APPLICATIONS TO GEOMETRY
8.2   DENSITY AND CENTER OF MASS
8.3   APPLICATIONS TO PHYSICS
8.4   APPLICATIONS TO ECONOMICS
REVIEW PROBLEMS
PROJECTS

FOCUS ON MODELING
DISTRIBUTION FUNCTIONS
PROBABILITY AND MORE ON DISTRIBUTIONS

9.  APPROXIMATIONS AND SERIES
9.1   TAYLOR POLYNOMIALS AND SERIES
9.2   CONVERGENCE OF SERIES
9.3   FINDING AND USING TAYLOR SERIES
9.4   GEOMETRIC SERIES
9.5   FOURIER SERIES
REVIEW PROBLEMS
PROJECTS

FOCUS ON THEORY
CONVERGENCE THEOREMS
THE ERROR IN TAYLOR APPROXIMATIONS

10.  DIFFERENTIAL EQUATIONS
10.1  WHAT IS A DIFFERENTIAL EQUATION?
10.2   SLOPE FIELDS
10.3   EULER'S METHOD
10.4   SEPARATION OF VARIABLES
10.5   GROWTH AND DECAY
10.6   APPLICATIONS AND MODELING
10.7   MODELS OF POPULATION GROWTH
10.8   SECOND-ORDER DIFFERENTIAL EQUATIONS:  OSCILLATIONS
10.9   LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS
REVIEW PROBLEMS
PROJECTS

11.  FUNCTIONS OF SEVERAL VARIABLES
11.1   FUNCTIONS OF TWO VARIABLES
11.2   A TOUR OF THREE-DIMENSIONAL SPACE
11.3   GRAPHS OF FUNCTIONS OF TWO VARIABLES
11.4   CONTOUR DIAGRAMS
11.5   LINEAR FUNCTIONS
11.6   FUNCTIONS OF MORE THAN TWO VARIABLES
REVIEW PROBLEMS
PROJECTS

FOCUS ON THEORY
LIMITS AND CONTINUITY

12.  A FUNDAMENTAL TOOL:  VECTORS
12.1   DISPLACEMENT VECTORS
12.2   VECTORS IN GENERAL
12.3   THE DOT PRODUCT
12.4   THE CROSS PRODUCT
REVIEW PROBLEMS
PROJECTS

13.  DIFFERENTIATING FUNCTIONS OF MANY VARIABLES
13.1   THE PARTIAL DERIVATIVE
13.2   COMPUTING PARTIAL DERIVATIVES ALGEBRAICALLY
13.3   LOCAL LINEARITY AND THE DIFFERENTIAL
13.4   GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE
13.5   GRADIENTS AND DIRECTIONAL DERIVATIVES IN SPACE
13.6   THE CHAIN RULE
13.7   SECOND ORDER PARTIAL DERIVATIVES
13.8   TAYLOR APPROXIMATIONS
REVIEW PROBLEMS
PROJECTS

FOCUS ON THEORY
DIFFERENTIABILITY

14.  OPTIMIZATION:  LOCAL AND GLOBAL EXTREMA
14.1   LOCAL EXTREMA
14.2   GLOBAL EXTREMA: UNCONSTRAINED OPTIMIZATION
14.3   CONSTRAINED OPTIMIZATION:  LAGRANGE MULTIPLIERS
REVIEW PROBLEMS
PROJECTS

FOCUS ON MODELING
THE LAGRANGIAN AND ITS INTERPRETATION

15.  INTEGRATING FUNCTIONS OF MANY VARIABLES
15.1   THE DEFINITE INTEGRAL OF A FUNCTION OF TWO VARIABLES
15.2   ITERATED INTEGRALS
15.3   TRIPLE INTEGRALS
15.4   DOUBLE INTEGRALS IN POLAR COORDINATES
15.5   INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
15.6   APPLICATIONS OF INTEGRATION TO PROBABILITY
REVIEW PROBLEMS
PROJECTS

FOCUS ON THEORY
CHANGE OF VARIABLES IN A MULTIPLE INTEGRAL

16.  PARAMETERIZED CURVES
16.1   PARAMETERIZED CURVES
16.2   MOTION, VELOCITY, AND ACCELERATION
REVIEW PROBLEMS
PROJECTS

17.  VECTOR FIELDS
17.1   VECTOR FIELDS
17.2   THE FLOW OF A VECTOR FIELD
REVIEW PROBLEMS

18.  LINE INTEGRALS
18.1   THE IDEA OF A LINE INTEGRAL
18.2   COMPUTING LINE INTEGRALS OVER PARAMETERIZED CURVES
18.3   GRADIENT FIELDS AND PATH-INDEPENDENT FIELDS
18.4   PATH-DEPENDENT VECTOR FIELDS AND GREEN'S THEOREM
REVIEW PROBLEMS
PROJECTS

FOCUS ON THEORY
PROOF OF GREEN'S THEOREM

19.  FLUX INTEGRALS
19.1   THE IDEA OF A FLUX INTEGRAL
19.2   FLUX INTEGRALS FOR GRAPHS, CYLINDERS, AND SPHERES
REVIEW PROBLEMS
PROJECTS

20.  CALCULUS OF VECTOR FIELDS
20.1   THE DIVERGENCE OF A VECTOR FIELD
20.2   THE DIVERGENCE THEOREM
20.3   THE CURL OF A VECTOR FIELD
20.4   STOKES' THEOREM
REVIEW PROBLEMS
PROJECTS

FOCUS ON THEORY
THE THREE FUNDAMENTAL THEOREMS

APPENDIX
A.  POLAR COORDINATES
B.  COMPLEX NUMBERS
C.  DETERMINANTS
D.  PROJECTS
ANSWERS TO OFF NUMBER PROBLEMS
INDEX