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Publisher:

W. H. Freeman

Publication Date:

2008

Number of Pages:

1027

Format:

Hardcover

Price:

172.20

ISBN:

9780716769118

Category:

Textbook

[Reviewed by , on ]

Allen Stenger

06/14/2007

This is an extremely conventional calculus book. Its claims to fame are that it has especially clear explanations, and that, because of thorough checking, it has few errors.

Because this text is so conventional it generally does not respond to any of the criticisms of the calculus reform movement. There is quite a lot of attention to symbolic integration techniques. Technology (graphing calculators and computer algebra systems) appears only in the exercises and not in the body. There are no student projects or writing exercises. The applications are all conventional, although many of them are improved by what appears to be real numerical data (there are no sources given for the data, so it's realistic but I'm not certain it's real).

The book does appear to be exceptionally error-free for a first printing. Richard Feynman's name is consistently misspelled. One of the infinite series exercises has an incorrect exponent. The book is generally very careful in its proofs. Most books give incorrect proofs of the chain rule, forgetting that there may be a division by zero in the process. This book gives a correct proof (although only in the exercises). On the other hand it falls down on the Bolzano-Weierstrass theorem and the least upper bound property — the given proofs are incorrect.

Some terminology is used in a non-standard way. To most people, f << g means the same as f=O(g), but here it means f=o(g). The book speaks of a "root" of a function (most people speak of roots of equations and zeroes of functions). Several exercises ask students to "prove" something using a numerical calculation in a CAS. These exercises treat the CAS as an oracle, not mentioning that numerical results are approximate, and that CASs, like other computer programs, sometimes make mistakes.

I like the numerous *Assumptions Matter* sections that explore what happens if the hypotheses are not satisfied. There are *Historical Perspective* sections scattered through the book; these don't really advance the exposition but they're interesting and may capture the student's attention.

But now we come to the key question: Given that there are already thousands of calculus books in print, is it valuable to have a new calculus book that is just like nearly all of them, except that it is has clearer explanations and fewer errors? It's hard to make this sound exciting, and in fact I am not excited by it. Most calculus books are deadly dull, and even though this one has lots of pretty pictures and interesting sidebars I still had a hard time getting through it. I would have liked it much better if it had addressed some of the issues raised by the reform movement. Still, the present book is very well done, and if you love traditional calculus books or for some reason you are prohibited from making any innovations in your calculus course, this may be the book for you!

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

**Chapter 1 PRECALCULUS REVIEW
** 1.1 Real Numbers, Functions, and Graphs

1.2 Linear and Quadratic Functions

1.3 The Basic Classes of Functions

1.4 Trigonometric Functions

1.5 Technology: Calculators and Computers

2.2 Limits: A Numerical and Graphical Approach

2.3 Basic Limit Laws

2.4 Limits and Continuity

2.5 Evaluating Limits Algebraically

2.6 Trigonometric Limits

2.7 Intermediate Value Theorem

2.8 The Formal Definition of a Limit

3.2 The Derivative as a Function

3.3 Product and Quotient Rules

3.4 Rates of Change

3.5 Higher Derivatives

3.6 Trigonometric Functions

3.7 The Chain Rule

3.8 Implicit Differentiation

3.9 Related Rates

4.2 Extreme Values

4.3 The Mean Value Theorem and Monotonicity

4.4 The Shape of a Graph

4.5 Graph Sketching and Asymptotes

4.6 Applied Optimization

4.7 Newton’s Method

4.8 Antiderivatives

5.2 The Definite Integral

5.3 The Fundamental Theorem of Calculus, Part I

5.4 The Fundamental Theorem of Calculus, Part II

5.5 Net or Total Change as the Integral of a Rate

5.6 Substitution Method

6.2 Setting Up Integrals: Volume, Density, Average Value

6.3 Volumes of Revolution

6.4 The Method of Cylindrical Shells

6.5 Work and Energy

7.2 Inverse Functions

7.3 Logarithms and their Derivatives

7.4 Exponential Growth and Decay

7.5 Compound Interest and Present Value

7.6 Models Involving y’= k(y-b)

7.7 L’Hoˆpital’s Rule

7.8 Inverse Trigonometric Functions

7.9 Hyperbolic Functions

8.2 Integration by Parts

8.3 Trigonometric Integrals

8.4 Trigonometric Substitution

8.5 The Method of Partial Fractions

8.6 Improper Integrals

9.2 Fluid Pressure and Force

9.3 Center of Mass

9.4 Taylor Polynomials

10.2 Graphical and Numerical Methods

10.3 The Logistic Equation

10.4 First-Order Linear Equations

11.2 Summing an Infinite Series

11.3 Convergence of Series with Positive Terms

11.4 Absolute and Conditional Convergence

11.5 The Ratio and Root Tests

11.6 Power Series

11.7 Taylor Series

12.2 Arc Length and Speed

12.3 Polar Coordinates

12.4 Area and Arc Length in Polar Coordinates

12.5 Conic Sections

13.2 Vectors in Three Dimensions

13.3 Dot Product and the Angle Between Two Vectors

13.4 The Cross Product

13.5 Planes in Three-Space

13.6 Survey of Quadric Surfaces

13.7 Cylindrical and Spherical Coordinates

14.2 Calculus of Vector-Valued Functions

14.3 Arc Length and Speed

14.4 Curvature

14.5 Motion in Three-Space

14.6 Planetary Motion According to Kepler and Newton

15.2 Limits and Continuity in Several Variables

15.3 Partial Derivatives

15.4 Differentiability, Linear Approximation,and Tangent Planes

15.5 The Gradient and Directional Derivatives

15.6 The Chain Rule

15.7 Optimization in Several Variables

15.8 Lagrange Multipliers: Optimizing with a Constraint

16.2 Double Integrals over More General Regions

16.3 Triple Integrals

16.4 Integration in Polar, Cylindrical, and Spherical Coordinates

16.5 Change of Variables

17.2 Line Integrals

17.3 Conservative Vector Fields

17.4 Parametrized Surfaces and Surface Integrals

17.5 Surface Integrals of Vector Fields

18.2 Stokes’ Theorem

18.3 Divergence Theorem

B. Properties of Real Numbers C. Mathematical Induction

and the BinomialTheorem D. Additional Proofs of Theorems

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