# 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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