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Publisher:

W. H. Freeman

Publication Date:

2013

Number of Pages:

1115

Format:

Hardcover

Price:

129.95

ISBN:

978-1-4292-4186-1

Category:

Textbook

[Reviewed by , on ]

Miklós Bóna

03/30/2013

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*

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