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Calculus

Dale Varberg, Edwin J. Purcell, and Steven E. Rigdon
Publisher: 
Prentice Hall
Publication Date: 
2007
Number of Pages: 
774
Format: 
Hardcover
Edition: 
9
Price: 
109.33
ISBN: 
0131469682
Category: 
Textbook
[Reviewed by
Fernando Q. Gouvêa
, on
03/25/2006
]

In the introduction, the authors say that this ninth edition of their book "continues to be the briefest of all the successful mainstream calculus texts," thereby laying claim to both success and brevity. I guess so. At 774 pages, this doesn't strike me as brief, really. But anything briefer must either not be successful or not be mainstream.

The table of contents is pretty much the standard one, making this a rather traditional text. It even includes a treatment of conic sections and of Kepler's Laws. One innovation is the inclusion of sets of problems that sit in between two chapters, reviewing what has just been done and attempting to preview what comes next. This is a pretty good idea, though I don't think the authors have actually made the most of it.

The production values are more modest than in many of the flashier books, which I actually prefer. (Only one color is used, for example.) Less good is the page layout, which seems quite cluttered to me, perhaps in the service of "brevity."

The approach is fairly theoretical, with many proofs and careful statements of theorems and definitions. On the other hand, it's unclear whether the problems follow through on this. In the section giving the definition of the derivative, for example, the problems begin (as in all sections of this book) with a "Concepts Review". But what is actually reviewed is the actual wording of the definition and of the theorem that differentiability implies continuity. This is a very superficial way of testing whether concepts have been understood. The problems that follow, for the most part, don't probe for that understanding either. If we assume that this book will be used, as most calculus texts seem to be, mainly as a source of problems, this seems like a lost opportunity.

So: a mostly traditional book, with the real virtue of brevity but otherwise not too imaginative.


Fernando Q. Gouvêa is professor of mathematics at Colby College, where he has taught second-semester calculus more times than he can count.

 

0 PRELIMINARIES

0.1

Real Numbers, Logic and Estimation

0.2

Inequalities and Absolute Values

0.3

The Rectangular Coordinate System

0.4

Graphs of Equations

0.5

Functions and Their Graphs

0.6

Operations on Functions

0.7

The Trigonometric Functions

 

1 LIMITS

1.1

Introduction to Limits

1.2

Rigorous Study of Limits

1.3

Limit Theorems

1.4

Limits Involving Trigonometric Functions

1.5

Limits at Infinity, Infinite Limits

1.6

Continuity of Functions

1.7

Chapter Review

 

2 THE DERIVATIVE

2.1

Two Problems with One Theme

2.2

The Derivative

2.3

Rules for Finding Derivatives

2.4

Derivatives of Trigonometric Functions

2.5

The Chain Rule

2.6

Higher-Order Derivatives

2.7

Implicit Differentiation

2.8

Related Rates

2.9

Differentials and Approximations

2.10

Chapter Review

 

3 APPLICATIONS OF THE DERIVATIVE

3.1

Maxima and Minima

3.2

Monotonicity and Concavity

3.3

Local Extrema and Extrema on Open Intervals

3.4

Graphing Functions Using Calculus

3.6

The Mean Value Theorem for Derivatives

3.7

Solving Equations Numerically

3.8

Antiderivatives

3.9

Introduction to Differential Equations

 

4 THE DEFINITE INTEGRAL

4.1

Introduction to Area

4.2

The Definite Integral

4.3

The 1st Fundamental Theorem of Calculus

4.4

The 2nd Fundamental Theorem of Calculus

and the Method of Substitution

4.5

The Mean Value Theorem for Integrals & the Use of Symmetry

4.6

Numerical Integration

4.7

Chapter Review

 

5 APPLICATIONS OF THE INTEGRAL

5.1

The Area of a Plane Region

5.2

Volumes of Solids: Slabs, Disks, Washers

5.3

Volumes of Solids of Revolution: Shells

5.4

Length of a Plane Curve

5.5

Work and Fluid Pressure

5.6

Moments, Center of Mass

5.7

Probability and Random Variables

5.8

Chapter Review

 

6 TRANSCENDENTAL FUNCTIONS

6.1

The Natural Logarithm Function

6.2

Inverse Functions and Their Derivatives

6.3

The Natural Exponential Function

6.4

General Exponential & Logarithmic Functions

6.5

Exponential Growth and Decay

6.6

First-Order Linear Differential Equations

6.7

Approximations for Differential Equations

6.8

Inverse Trig Functions & Their Derivatives

6.9

The Hyperbolic Functions & Their Inverses

6.10

Chapter Review

 

7 TECHNIQUES OF INTEGRATION

7.1

Basic Integration Rules

7.2

Integration by Parts

7.3

Some Trigonometric Integrals

7.4

Rationalizing Substitutions

7.5

The Method of Partial Fractions

7.6

Strategies for Integration

7.7

Chapter Review

 

8 INDETERMINATE FORMS &  IMPROPER INTEGRALS

8.1

Indeterminate Forms of Type 0/0

8.2

Other Indeterminate Forms

8.3

Improper Integrals: Infinite Limits of Integration

8.4

Improper Integrals: Infinite Integrands

8.5

Chapter Review

 

9 INFINITE SERIES

9.1

Infinite Sequences

9.2

Infinite Series

9.3

Positive Series: The Integral Test

9.4

Positive Series: Other Tests

9.5

Alternating Series, Absolute Convergence,

and Conditional Convergence

9.6

Power Series

9.7

Operations on Power Series

9.8

Taylor and Maclaurin Series

9.9

The Taylor Approximation to a Function

9.10

Chapter Review

 

10 CONICS AND POLAR COORDINATES

10.1

The Parabola

10.2

Ellipses and Hyperbolas

10.3

Translation and Rotation of Axes

10.4

Parametric Representation of Curves

10.5

The Polar Coordinate System

10.6

Graphs of Polar Equations

10.7

Calculus in Polar Coordinates

10.8

Chapter Review

 

11 GEOMETRY IN SPACE, VECTORS

11.1

Cartesian Coordinates in Three-Space

11.2

Vectors

11.3

The Dot Product

11.4

The Cross Product

11.5

Vector Valued Functions & Curvilinear Motion

11.6

Lines in Three-Space

11.7

Curvature and Components of Acceleration

11.8

Surfaces in Three Space

11.9

Cylindrical and Spherical Coordinates

11.10

Chapter Review

 

12 DERIVATIVES OF FUNCTIONS OF TWO OR MORE VARIABLES

12.1

Functions of Two or More Variables

12.2

Partial Derivatives

12.3

Limits and Continuity

12.4

Differentiability

12.5

Directional Derivatives and Gradients

12.6

The Chain Rule

12.7

Tangent Planes, Approximations

12.8

Maxima and Minima

12.9

Lagrange Multipliers

12.10

Chapter Review

 

13 MULTIPLE INTEGRATION

13.1

Double Integrals over Rectangles

13.2

Iterated Integrals

13.3

Double Integrals over Nonrectangular Regions

13.4

Double Integrals in Polar Coordinates

13.5

Applications of Double Integrals

13.6

Surface Area

13.7

Triple Integrals (Cartesian Coordinates)

13.8

Triple Integrals (Cyl & Sph Coordinates)

13.9

Change of Variables in Multiple Integrals

13.1

Chapter Review

 

14 VECTOR CALCULUS

14.1

Vector Fields

14.2

Line Integrals

14.3

Independence of Path

14.4

Green's Theorem in the Plane

14.5

Surface Integrals

14.6

Gauss's Divergence Theorem

14.7

Stokes's Theorem

14.8

Chapter Review

 

APPENDIX

A.1

Mathematical Induction

A.2

Proofs of Several Theorems

A.3

A Backward Look

 

 
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