You are here

Calculus

Publisher: 
W. H. Freeman
Number of Pages: 
1027
Price: 
172.20
ISBN: 
9780716769118

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.

Date Received: 
Tuesday, February 13, 2007
Reviewable: 
Yes
Include In BLL Rating: 
No
Jon Rogawski
Publication Date: 
2008
Format: 
Hardcover
Audience: 
Category: 
Textbook
Allen Stenger
06/14/2007

  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
    
  Chapter 2 LIMITS
    2.1 Limits, Rates of Change, and Tangent Lines
    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
    
  Chapter 3 DIFFERENTIATION
    3.1 Definition of the Derivative
    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
    
  Chapter 4 APPLICATIONS OF THE DERIVATIVE
    4.1 Linear Approximation and Applications
    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
    
  Chapter 5 THE INTEGRAL
    5.1 Approximating and Computing Area
    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
    
  Chapter 6 APPLICATIONS OF THE INTEGRAL
    6.1 Area Between Two Curves
    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
    
  Chapter 7 EXPONENTIAL FUNCTIONS
    7.1 Derivative of f(x)=b^x and the Number e
    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
    
  Chapter 8 TECHNIQUES OF INTEGRATION
    8.1 Numerical Integration
    8.2 Integration by Parts
    8.3 Trigonometric Integrals
    8.4 Trigonometric Substitution
    8.5 The Method of Partial Fractions
    8.6 Improper Integrals
    
  Chapter 9 FURTHER APPLICATIONS OF THE INTEGRAGAL TAYLOR POLYNOMIALS
    9.1 Arc Length and Surface Area
    9.2 Fluid Pressure and Force
    9.3 Center of Mass
    9.4 Taylor Polynomials
    
  Chapter 10 INTRODUCTION TO DIFFERENTIAL EQUATIONS
    10.1 Solving Differential Equations
    10.2 Graphical and Numerical Methods
    10.3 The Logistic Equation
    10.4 First-Order Linear Equations
    
  Chapter 11 INFINITE SERIES
    11.1 Sequences
    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
    
  Chapter 12 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS
    12.1 Parametric Equations
    12.2 Arc Length and Speed
    12.3 Polar Coordinates
    12.4 Area and Arc Length in Polar Coordinates
    12.5 Conic Sections
    
  Chapter 13 VECTOR GEOMETRY
    13.1 Vectors in the Plane
    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
    
  Chapter 14 CALCULUS OF VECTOR-VALUED FUNCTIONS
    14.1 Vector-Valued Functions
    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
    
  Chapter 15 DIFFERENTIATION IN SEVERAL VARIABLES
    15.1 Functions of Two or More Variables
    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
    
  Chapter 16 MULTIPLE INTEGRATION
    16.1 Integration in Several Variables
    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
    
  Chapter 17 LINE AND SURFACE INTEGRALS
    17.1 Vector Fields
    17.2 Line Integrals
    17.3 Conservative Vector Fields
    17.4 Parametrized Surfaces and Surface Integrals
    17.5 Surface Integrals of Vector Fields
    
  Chapter 18 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS
    18.1 Green’s Theorem
    18.2 Stokes’ Theorem
    18.3 Divergence Theorem
    
  APPENDICES
    A. The Language of Mathematics
    B. Properties of Real Numbers C. Mathematical Induction
    and the BinomialTheorem D. Additional Proofs of Theorems
    
  ANSWERS TO ODD-NUMBERED EXERCISES

Publish Book: 
Modify Date: 
Thursday, June 14, 2007

Dummy View - NOT TO BE DELETED