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Calculus: Early Transcendentals

Publisher: 
Pearson/Prentice Hall
Number of Pages: 
790
Price: 
109.33
ISBN: 
0132273462

This is the "early transcendentals" version of a text that was reviewed here earlier. Most of what was said in that review applies to this book also. The main difference is that the logarithm and exponential functions are introduced much earlier in this version of the text.

The need for an "early transcendentals" option is a consequence of the widespread adoption of the idea (first proposed, I believe, by Felix Klein in his Elementary Mathematics from an Advanced Standpoint, where he argues that it be used in the schools ) that the natural logarithm is best introduced as the integral of 1/x and that the exponential is best defined as the inverse of the natural logarithm.

Adopting Klein's suggestion was always a rather strange decision in the context of the average undergraduate calculus text. After all, no one felt the need to include a rigorous definition of the sine and cosine functions. Giving such a definition also requires doing some (rather hard) analysis, as one can see in Spivak's Calculus and in most standard real analysis texts. Furthermore, undergraduates arrive in college already "knowing" about these functions; it seems reasonable to simply recall their properties and use them, as we do for the trigonometric functions.

The approach taken in this book is less radical than that. The authors define ax for rational x in the usual way (without bothering to prove the existence of n-th roots, of course), then extend the definition to irrational exponents using a limit argument (with at least some proofs). They then define the logarithm as the inverse function, and finally use the limit of (1+r/n)n to introduce the "natural" exponential function. (At this point of the text, not a word is said to justify that adjective!) The derivative of the logarithm function is obtained first, then the inverse function theorem is used to find the derivative of the exponential function.

This approach does have the advantage of introducing a serious example of the use of limits into what is usually a dreary chapter. (Until we meet the derivative, why do we care what the limit is?) I believe it is an improvement on the "late transcendentals" approach. But in the end I still have my doubts whether students are ready for this kind of analysis so early in the course.

For what it's worth, I would simply take for granted the existence of the logarithm and exponential functions and their basic properties. Then, using the definition of the derivative, it's fairly easy to prove that the derivative of ax is some constant times ax. Choosing the right value for a makes that constant equal to 1, which explains what is "natural" about ex . An analogous argument explains why we must use radian measure if we want the derivative of sin(x) to be cos(x) rather than a constant times cos(x). This seems both simpler and more illuminating, even if it hides the inherent trickiness of defining these functions. There's time for that in the analysis course.


Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME.

Date Received: 
Tuesday, May 30, 2006
Reviewable: 
Include In BLL Rating: 
Dale Varberg, Edwin J. Purcell, and Steven E. Rigdon
Publication Date: 
2007
Format: 
Hardcover
Audience: 
Category: 
Textbook
Tags: 
Fernando Q. Gouêa
05/30/2006

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

1

 

FUNCTIONS

 

1.1

Functions and Their Graphs

 

1.2

Operations on Functions

 

1.3

Exponential and Logarithmic Functions

 

1.4

The Trigonometric Functions & Their Inverses

 

1.5

Chapter Review

2

 

LIMITS

 

2.1

Introduction to Limits

 

2.2

Rigorous Study of Limits

 

2.3

Limit Theorems

 

2.4

Limits Involving Transcendental Functions

 

2.5

Limits at Infinity, Infinite Limits

 

2.6

Continuity of Functions

 

2.7

Chapter Review

3

 

THE DERIVATIVE

 

3.1

Two Problems with One Theme

 

3.2

The Derivative

 

3.3

Rules for Finding Derivatives

 

3.4

Derivatives of Trigonometric Functions

 

3.5

The Chain Rule

 

3.6

Higher-Order Derivatives

 

3.7

Implicit Differentiation

 

3.8

Related Rates

 

3.9

Differentials and Approximations

 

3.10

Chapter Review

4

 

APPLICATIONS OF THE DERIVATIVE

 

4.1

Maxima and Minima

 

4.2

Monotonicity and Concavity

 

4.3

Local Extrema and Extrema on Open Intervals

 

4.4

Graphing Functions Using Calculus

 

4.5

The Mean Value Theorem for Derivatives

 

4.6

Solving Equations Numerically

 

4.7

Antiderivatives

 

4.8

Introduction to Differential Equations

5

 

THE DEFINITE INTEGRAL

 

5.1

Introduction to Area

 

5.2

The Definite Integral

 

5.3

The 1st Fundamental Theorem of Calculus

 

5.4

The 2nd Fundamental Theorem of Calculus

 

 

and the Method of Substitution

 

5.5

The Mean Value Theorem for Integrals & the Use of Symmetry

 

5.6

Numerical Integration

 

5.7

Chapter Review

6

 

APPLICATIONS OF THE INTEGRAL

 

6.1

The Area of a Plane Region

 

6.2

Volumes of Solids: Slabs, Disks, Washers

 

6.3

Volumes of Solids of Revolution: Shells

 

6.4

Length of a Plane Curve

 

6.5

Work and Fluid Pressure

 

6.6

Moments, Center of Mass

 

6.8

Probability and Random Variables

 

6.8

Chapter Review

7

 

TECHNIQUES OF INTEGRATION &

 

 

DIFFERENTIAL EQUATIONS

 

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

Growth and Decay

 

7.8

First-Order Linear Differential Equations

 

7.9

Approximations for Differential Equations

 

7.10

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

Publish Book: 
Modify Date: 
Tuesday, May 30, 2006

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