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Calculus: Concepts & Connections

Robert T. Smith and Roland B. Minton
Publisher: 
McGraw-Hill
Publication Date: 
2006
Number of Pages: 
1104
Format: 
Hardcover
Price: 
132.81
ISBN: 
0-07-282623-1
Category: 
Textbook
[Reviewed by
William Satzer
, on
06/11/2005
]

Calculus: Concepts and Connections is a readable introduction to the basic topics of single and multivariable calculus from the basic idea of limits up through Stokes' Theorem. It is a textbook clearly aimed at the "average" student. The main difference between this and other comparable textbooks such as Stewart's Calculus is that explanations of concepts and examples are more extensive and more explicit attempts are made to establish the connections between concepts. The standard topics are treated in more or less the usual order.

The authors have obviously taken pains to develop good exercise sets. Typically, each chapter has writing exercises, a collection of fairly routine problems, exercises designed to use a graphing calculator or computer algebra system, and exploratory problems. The latter are intended to be more challenging and to provoke a deeper level of understanding. A nice feature in the text is the use of an icon to identify pitfalls arising from injudicious use of calculators or computer algebra systems.

There are well over two hundred explicit examples of applications in the text or in exercises; these fall into the general categories of biology, chemistry, economics, physiology, engineering, physics and sports. While these are not especially deep applications, they generally have enough meat on them to be interesting.

Intuitive explanations and arguments of plausibility are usually favored over proofs, though there are a variety of proofs of varying levels of generality and rigor. In addition, there is an appendix with more rigorous proofs of some results.

This would be an appealing self-study text as well as a good choice for a class of average ability. It would probably seem too slow moving for strong students.

Bill Satzer ([email protected]) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics. 

2 pp. -- Endpaper
Add 3 pp. -- Index of Applications
Add 5 pp. -- Preface
Add 6 pp. -- Guided Tour
Add 1 pg. -- To the Student
0. Preliminaries
Add 2 pp. -- Introduction
Add 14 pp. -- 0.1 Polynomials and Rational Functions
Add 3 pp. -- Exercises 0.1
Add 8 pp. -- 0.2 Graphing Calculators and Computer Algebra Systems
Add 2 pp. -- Exercises 0.2
Add 5 pp. -- 0.3 Inverse Functions
Add 2 pp. -- Exercises 0.3
Add 10 pp. -- 0.4 Trigonometric and Inverse Trigonometric Functions
Add 4 pp. -- Exercises 0.4
Add 11 pp. -- 0.5 Exponential and Logarithmic Functions
Add 3 pp. -- Exercises 0.5
Add 7 pp. -- 0.6 Transformations of Functions
Add 2 pp. -- Exercises 0.6
Add 13 pp. -- 0.7 Parametric Equations and Polar Coordinates
Add 4 pp. -- Exercises 0.7
Add 5 pp. -- Review Exercises
1. Limits and Continuity
Add 2 pp. -- Introduction
Add 5 pp. -- 1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve
Add 2 pp. -- Exercises 1.1
Add 7 pp. -- 1.2 The Concept of Limit
Add 3 pp. -- Exercise 1.2
Add 10 pp. -- 1.3 Computation of Limits
Add 3 pp. -- Exercises 1.3
Add 9 pp. -- 1.4 Continuity and Its Consequences
Add 5 pp. -- Exercises 1.4
Add 9 pp. -- 1.5 Limits Involving Infinity
Add 4 pp. -- Exercises 1.5
Add 7 pp. -- 1.6 Limits and Loss-Of-Significance Errors
Add 2 pp. -- Exercises 1.6
Add 4 pp. -- Review Exercises
2. Differentiation
Add 2 pp. -- Introduction
Add 9 pp. -- 2.1 Tangent Lines and Velocity
Add 4 pp. -- Exercises 2.1
Add 8 pp. -- 2.2 The Derivative
Add 5 pp. -- Exercises 2.2
Add 7 pp. -- 2.3 Computation of Derivatives: The Power Rule
Add 4 pp. -- Exercises 2.3
Add 6 pp. -- 2.4 The Product and Quotient Rules
Add 4 pp. -- Exercises 2.4
Add 7 pp. -- 2.5 The Chain Rule
Add 3 pp. -- Exercises 2.5
Add 7 pp. -- 2.6 Derivatives of Trigonometric Functions
Add 3 pp. -- Exercises 2.6
Add 8 pp. -- 2.7 Derivatives of Exponential and Logarithmic Functions
Add 3 pp. -- Exercises 2.7
Add 8 pp. -- 2.8 Implicit Differentiation and Inverse Trigonometric Functions
Add 3 pp. -- Exercises 2.8
Add 6 pp. -- 2.9 The Mean Value Theorem
Add 3 pp. -- Exercises 2.9
Add 5 pp. -- Review Exercise
3. Applications of Differentiation
Add 2 pp. -- Introduction
Add 9 pp. -- 3.1 Linear Approximations and Newton?s Method
Add 5 pp. -- Exercises 3.1
Add 9 pp. -- 3.2 Indeterminate Forms and L?H¢pital?s Rule
Add 3 pp. -- Exercises 3.2
Add 9 pp. -- 3.3 Maximum and Minimum Values
Add 4 pp. -- Exercises 3.3
Add 9 pp. -- 3.4 Increasing and Decreasing Functions
Add 3 pp. -- Exercises 3.4
Add 12 pp. -- 3.5 Concavity and Overview of Curve Sketching
Add 5 pp. -- Exercises 3.5
Add 8 pp. -- 3.6 Optimization
Add 5 pp. -- Exercises 3.6
Add 8 pp. -- 3.7 Rates of Change in Economics and the Sciences
Add 4 pp. -- Exercises 3.7
Add 6 pp. -- 3.8 Related Rates and Parametric Equations
Add 4 pp. -- Exercises 3.8
Add 3 pp. -- Review Exercises
4. Integration
Add 2 pp. -- Introduction
Add 10 pp. -- 4.1 Area Under a Curve
Add 3 pp. -- Exercises 4.1
Add 11 pp. -- 4.2 The Definite Integral
Add 4 pp. -- Exercises 4.2
Add 9 pp. -- 4.3 Antiderivatives
Add 3 pp. -- Exercises 4.3
Add 8 pp. -- 4.4 The Fundamental Theorem of Calculus
Add 3 pp. -- Exercises 4.4
Add 6 pp. -- 4.5 Integration by Substitution
Add 3 pp. -- Exercises 4.5
Add 6 pp. -- 4.6 Integration by Parts
Add 2 pp. -- Exercises 4.6
Add 11 pp. -- 4.7 Other Techniques of Integration
Add 3 pp. -- Exercises 4.7
Add 8 pp. -- 4.8 Integration Tables and Computer Algebra Systems
Add 2 pp. -- Exercises 4.8
Add 11 pp. -- 4.9 Numerical Integration
Add 4 pp. -- Exercises 4.9
Add 12 pp. -- 4.10 Improper Integrals
Add 3 pp. -- Exercises 4.10
Add 4 pp. -- Review Exercises
5. Applications of the Definite Integral
Add 1 pg. -- Introduction
Add 7 pp. -- 5.1 Area of a Plane Region
Add 4 pp. -- Exercises 5.1
Add 16 pp. -- 5.2 Volume
Add 4 pp. -- Exercises 5.2

Add 7 pp. -- 5.3 Arc Length and Surface Area
Add 3 pp. -- Exercises 5.3
Add 7 pp. -- 5.4 Projectile Motion
Add 5 pp. -- Exercises 5.4
Add 10 pp. -- 5.5 Applications of Integration to Physics and Engineering
Add 4 pp. -- Exercises 5.5
Add 8 pp. -- 5.6 Probability
Add 4 pp. -- Exercises 5.6
Add 5 pp. -- Review Exercises
6. Differential Equations
Add 2 pp. -- Introduction
Add 8 pp. -- 6.1 Growth and Decay Problems
Add 5 pp. -- Exercises 6.1
Add 8 pp. -- 6.2 Separable Differential Equations
Add 4 pp. -- Exercises 6.2
Add 10 pp. -- 6.3 Direction Fields and Euler?s Method
Add 4 pp. -- Exercises 6.3
Add 8 pp. -- 6.4 Second-Order Equations with Constant Coefficients
Add 3 pp. -- Exercises 6.4
Add 8 pp. -- 6.5 Nonhomogeneous Equations: Undetermined Coefficients
Add 4 pp. -- Exercises 6.5
Add 3 pp. -- Review Exercises
7. Infinite Series
Add 2 pp. -- Introduction
Add 10 pp. -- 7.1 Sequences of Real Numbers
Add 3 pp. -- Exercises 7.1
Add 8 pp. -- 7.2 Infinite Series
Add 3 pp. -- Exercises 7.2
Add 9 pp. -- 7.3 The Integral Test and Comparison Tests
Add 3 pp. -- Exercises 7.3
Add 6 pp. -- 7.4 Alternating Series
Add 3 pp. -- Exercises 7.4
Add 6 pp. -- 7.5 Absolute Convergence and the Ratio Test
Add 2 pp. -- Exercises 7.5
Add 7 pp. -- 7.6 Power Series
Add 3 pp. -- Exercises 7.6
Add 12 pp. -- 7.7 Taylor Series
Add 3 pp. -- Exercises 7.7
Add 7 pp. -- 7.8 Applications of Taylor Series
Add 2 pp. -- Exercises 7.8
Add 13 pp. -- 7.9 Fourier Series
Add 4 pp. -- Exercises 7.9
Add 10 pp. -- 7.10 Using Series to Solve Differential Equations
Add 2 pp. -- Exercises 7.10
Add 4 pp. -- Review Exercises
8. Vectors and the Geometry of Space
Add 2 pp. -- Introduction
Add 9 pp. -- 8.1 Vectors in the Plane
Add 3 pp. -- Exercises 8.1
Add 7 pp. -- 8.2 Vectors in Space
Add 3 pp. -- Exercises 8.2
Add 7 pp. -- 8.3 The Dot Product
Add 5 pp. -- Exercises 8.3
Add 10 pp. -- 8.4 The Cross Product
Add 3 pp. -- Exercises 8.4
Add 8 pp. -- 8.5 Lines and Planes in Space
Add 3 pp. -- Exercises 8.5
Add 11 pp. -- 8.6 Surfaces in Space
Add 3 pp. -- Exercises 8.6
Add 3 pp. -- Review Exercises
9. Vector-Valued Functions
Add 1 pg. -- Introduction
Add 8 pp. -- 9.1 Vector-Valued Functions
Add 4 pp. -- Exercises 9.1
Add 10 pp. -- 9.2 The Calculus of Vector-Valued Functions
Add 2 pp. -- Exercises 9.2
Add 9 pp. -- 9.3 Motion in Space
Add 4 pp. -- Exercises 9.3
Add 7 pp. -- 9.4 Curvature
Add 2 pp. -- Exercises 9.4
Add 13 pp. -- 9.5 Tangent and Normal Vectors
Add 3 pp. -- Exercises 9.5
Add 4 pp. -- 9.6 Parametric Surfaces
Add 3 pp. -- Exercises 9.6
Add 4 pp. -- Review Exercises
10. Functions of Several Variables and Partial Differentiation
Add 1 pg. -- Introduction
Add 2 pp. -- 10.1 Functions of Several Variables
Add 8 pp. -- Exercises 10.1
Add 10 pp. -- 10.2 Limits and Continuity
Add 2 pp. -- Exercises 10.2
Add 9 pp. -- 10.3 Partial Derivatives
Add 5 pp. -- Exercises 10.3
Add 11 pp. -- 10.4 Tangent Planes and Linear Approximations
Add 4 pp. -- Exercises 10.4
Add 9 pp. -- 10.5 The Chain Rule
Add 3 pp. -- Exercises 10.5
Add 9 pp. -- 10.6 The Gradient and Directional Derivatives
Add 5 pp. -- Exercises 10.6
Add 12 pp. -- 10.7 Extrema of Functions of Several Variables
Add 4 pp. -- Exercises 10.7
Add 10 pp. -- 10.8 Constrained Optimization and Lagrange Multipliers
Add 3 pp. -- Exercises 10.8
Add 7 pp. -- Review Exercises
11. Multiple Integrals
Add 2 pp. -- Introduction
Add 14 pp. -- 11.1 Double Integrals
Add 4 pp. -- Exercises 11.1
Add 9 pp. -- 11.2 Area, Volume and Center of Mass
Add 4 pp. -- Exercises 11.2
Add 6 pp. -- 11.3 Double Integrals in Polar Coordinates
Add 3 pp. -- Exercises 11.3
Add 6 pp. -- 11.4 Surface Area
Add 2 pp. -- Exercises 11.4
Add 11 pp. -- 11.5 Triple Integrals
Add 3 pp. -- Exercises 11.5
Add 6 pp. -- 11.6 Cylindrical Coordinates

Add 2 pp. -- Exercises 11.6
Add 8 pp. -- 11.7 Spherical Coordinates
Add 3 pp. -- Exercises 11.7
Add 11 pp. -- 11.8 Change of Variables in Multiple Integrals
Add 3 pp. -- Exercises 11.8
Add 3 pp. -- Review Exercises
12. Vector Calculus
Add 1 pg. -- Introduction
Add 12 pp. -- 12.1 Vector Fields
Add 4 pp. -- Exercises 12.1
Add 7 pp. -- 12.2 Curl and Divergence
Add 3 pp. -- Exercises 12.2
Add 14 pp. -- 12.3 Line Integrals
Add 4 pp. -- Exercises 12.3
Add 7 pp. -- 12.4 Independence of Path and Conservative Vector Fields
Add 3 pp. -- Exercises 12.4
Add 9 pp. -- 12.5 Green?s Theorem
Add 2 pp. -- Exercises 12.5
Add 11 pp. -- 12.6 Surface Integrals
Add 3 pp. -- Exercises 12.6
Add 6 pp. -- 12.7 The Divergence Theorem
Add 3 pp. -- Exercises 12.7
Add 9 pp. -- 12.8 Stokes? Theorem
Add 2 pp. -- Exercises 12.8
Add 7 pp. -- 12.9 Applications of Vector Calculus
Add 2 pp. -- Exercises 12.9
Add 5 pp. -- Review Exercises
Back Matter
Add 4 pp. -- Appendix A: Additional Polar Graphs
Add 13 pp. -- Appendix B: Formal Definition of the Limit
Add 4 pp. -- Appendix C: Derivative of the Sine Function
Add 10 pp. -- Appendix D: The Natural Logarithm as an Integral
Add 7 pp. -- Appendix E: Conic Sections in Polar Coordinates
12 pp. -- Appendix F: Proofs of Selected Theorems
2 pp. -- Answers to Odd-Numbered Exercises Chapter 0
2 pp. -- Answers to Odd-Numbered Exercises Chapter 0 Review
Add 3 pp. -- Answers to Odd-Numbered Exercises chapter 1
Add 2 pp. -- Answers to Odd-Numbered Exercises Chapter 1 Review
Add 6 pp. -- Answers to Odd-Numbered Exercises chapter 2
Add 2 pp. -- Answers to Odd-Numbered Exercises Chapter 2 Review
Add 5 pp. -- Answers to Odd-Numbered Exercises chapter 3
Add 1 pg. -- Answers to Odd-Numbered Exercises Chapter 3 Review
Add 3 pp. -- Answers to Odd-Numbered Exercises chapter 4
Add 1 pg. -- Answers to Odd-Numbered Exercises Chapter 4 Review
Add 4 pp. -- Answers to Odd-Numbered Exercises chapter 5
Add 2 pp. -- Answers to Odd-Numbered Exercises chapter 5 Review
Add 4 pp. -- Answers to Odd-Numbered Exercises chapter 6
Add 1 pg. -- Answers to Odd-Numbered Exercises chapter 6 Review
Add 5 pp. -- Answers to Odd-Numbered Exercises Chapter 7
Add 2 pp. -- Answers to Odd-Numbered Exercises Chapter 7 Review
Add 5 pp. -- Answers to Odd-Numbered Exercises Chapter 8
Add 1 pg. -- Answers to Odd-Numbered Exercises Chapter 8 Review
Add 4 pp. -- Answers to Odd-Numbered Exercises Chapter 9
Add 2 pp. -- Answers to Odd-Numbered Exercises Chapter 10
Add 4 pp. -- Answers to Odd-Numbered Exercises Chapter 10 Review
Add 2 pp. -- Answers to Odd-Numbered Exercises Chapter 11
Add 3 pp. -- Answers to Odd-Numbered Exercises Chapter 11 Review
Add 2 pp. -- Answers to Odd-Numbered Exercises Chapter 12
Add 2 pp. -- Answers to Odd-Numbered Exercises Chapter 12 Review
Add 10 pp. -- Bibliography
Add 1 pg. -- Credits
Add 15 pp. -- Index
Add 4 pp. -- Derivative Formulas
Add 2 pp. -- Table of Integrals
Tags: 
Calculus