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Laura Taalman and Peter Kohn
W. H. Freeman
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
, on

The Calculus textbook market is crowded, and the books on that market are very similar to one another. It is completely normal for two books in that category to overlap by ninety percent or more in their coverage of topics. The task of the reviewer is therefore to discuss how that book at hand differs from the rest.

The pace of this book is slightly faster than that of a typical competing textbook. For instance, limits are introduced in the sixth section, not in the eighth as in a popular alternative. Accordingly, derivatives and integrals are also reached a little bit earlier. The book has only fourteen chapters, because it does not cover differential equations. Many colleges and universities will consider that as an advantage, since differential equations are typically taught in a different class, so including them in a calculus book is not necessary.

The most impressive part of the book is its exercises. Their number is comparable to that in competing textbooks, but the exercises of this book are much better organized, much more diverse, and much more useful for students than in the overwhelming majority of the competition. After each section, the exercises are divided into numerous subsections, such as Thinking Back, Concepts, Skills, Applications, Proofs, and Thinking Forward. The “Concepts” part often contains True or False questions testing students′ understanding of the notions just covered. This is completely absent in most competing textbooks, just as exercises that ask that the student prove something are absent. If you cannot teach from the book, it is still worth getting a copy for the exercises.

Miklós Bóna is Professor of Mathematics at the University of Florida.

Part I. Differential Calculus

0. Functions and Precalculus
0.1 Functions and Graphs
0.2 Operations, Transformations, and Inverses
0.3 Algebraic Functions
0.4 Exponential and Trigonometric Functions
0.5 Logic and Mathematical Thinking*
Chapter Review, Self-Test, and Capstones

1. Limits
1.1 An Intuitive Introduction to Limits
1.2 Formal Definition of Limit
1.3 Delta-Epsilon Proofs*
1.4 Continuity and Its Consequences
1.5 Limit Rules and Calculating Basic Limits
1.6 Infinite Limits and Indeterminate Forms
Chapter Review, Self-Test, and Capstones

2. Derivatives
2.1 An Intuitive Introduction to Derivatives
2.2 Formal Definition of the Derivative
2.3 Rules for Calculating Basic Derivatives
2.4 The Chain Rule and Implicit Differentiation
2.5 Derivatives of Exponential and Logarithmic Functions
2.6 Derivatives of Trigonometric and Hyperbolic Functions*
Chapter Review, Self-Test, and Capstones

3. Applications of the Derivative
3.1 The Mean Value Theorem
3.2 The First Derivative and Curve Sketching
3.3 The Second Derivative and Curve Sketching
3.4 Optimization
3.5 Related Rates
3.6 L’Hopital’s Rule
Chapter Review, Self-Test, and Capstones

Part II. Integral Calculus

4. Definite Integrals
4.1 Addition and Accumulation
4.2 Riemann Sums
4.3 Definite Integrals
4.4 Indefinite Integrals
4.5 The Fundamental Theorem of Calculus
4.6 Areas and Average Values
4.7 Functions Defined by Integrals
Chapter Review, Self-Test, and Capstones

5. Techniques of Integration
5.1 Integration by Substitution
5.2 Integration by Parts
5.3 Partial Fractions and Other Algebraic Techniques
5.4 Trigonometric Integrals
5.5 Trigonometric Substitution
5.6 Improper Integrals
5.7 Numerical Integration*
Chapter Review, Self-Test, and Capstones

6. Applications of Integration
6.1 Volumes By Slicing
6.2 Volumes By Shells
6.3 Arc Length and Surface Area
6.4 Real-World Applications of Integration
6.5 Differential Equations*
Chapter Review, Self-Test, and Capstones

Part III. Sequences and Series

7. Sequences and Series
7.1 Sequences
7.2 Limits of Sequence
7.3 Series
7.4 Introduction to Convergence Tests
7.5 Comparison Tests
7.6 The Ratio and Root Tests
7.7 Alternating Series
Chapter Review, Self-Test, and Capstones
8. Power Series
8.1 Power Series
8.2 Maclaurin Series and Taylor Series
8.3 Convergence of Power Series
8.4 Differentiating and Integrating Power Series
Chapter Review, Self-Test, and Capstones

Part IV. Vector Calculus

9. Parametric Equations, Polar Coordinates, and Conic Sections
9.1 Parametric Equations
9.2 Polar Coordinates
9.3 Graphing Polar Equations
9.4 Computing Arc Length and Area with Polar Functions
9.5 Conic Sections*
Chapter Review, Self-Test, and Capstones

10. Vectors
10.1 Cartesian Coordinates
10.2 Vectors
10.3 Dot Product
10.4 Cross Product
10.5 Lines in Three-Dimensional Space
10.6 Planes
Chapter Review, Self-Test, and Capstones

11. Vector Functions
11.1 Vector-Valued Functions
11.2 The Calculus of Vector Functions
11.3 Unit Tangent and Unit Normal Vectors
11.4 Arc Length Parametrizations and Curvature
11.5 Motion
Chapter Review, Self-Test, and Capstones

Part V. Multivariable Calculus

12. Multivariable Functions
12.1 Functions of Two and Three Variables
12.2 Open Sets, Closed Sets, Limits, and Continuity
12.3 Partial Derivatives
12.4 Directional Derivatives and Differentiability
12.5 The Chain Rule and the Gradient
12.6 Extreme Values
12.7 Lagrange Multipliers
Chapter Review, Self-Test, and Capstones

13. Double and Triple Integrals
13.1 Double Integrals over Rectangular Regions
13.2 Double Integrals over General Regions
13.3 Double Integrals in Polar Coordinates
13.4 Applications of Double Integrals
13.5 Triple Integrals
13.6 Integration with Cylindrical and Spherical Coordinates
13.7 Jacobians and Change of Variables
Chapter Review, Self-Test, and Capstones

14. Vector Analysis
14.1 Vector Fields
14.2 Line Integrals
14.3 Surfaces and Surface Integrals
14.4 Green’s Theorem
14.5 Stokes’ Theorem
14.6 The Divergence Theorem
Chapter Review, Self-Test, and Capstones