The vast majority of the topics found in our book are part of the standard calculus curriculum that has defined the mainstream for the last 30 years or so. We believe that this curriculum still has validity in terms of both mathematical precision and student learning. Nevertheless, we have made a small number of significant changes in the table of contents. Following find our own brief explanation of each chapter and its focus.
Chapter 0: Preliminaries - 0.1 The Real Numbers and the Cartesian Plane
- 0.2 Lines and Functions
- 0.3 Graphing Calculators and Computer Algebra Systems
- 0.4 Solving Equations
- 0.5 Trigonometric Functions
- 0.6 Exponential and Logarithmic Functions
- 0.7 Transformations of Functions
- 0.8 Preview of Calculus
Chapter 0 consists mostly of a review of background material. Depending on the focus of a specific course, instructors may decide to cover all, some or none of this material. This chapter focuses attention on those aspects of algebra and trigonometry that will be most useful to students as they progress through calculus. In particular, Chapter 0 contains a review of the basic properties of exponential, logarithmic, and trigonometric functions. Sections 0.3 and 0.4 can be used to familiarize students with the use of a graphing calculator or computer algebra system. Section 0.5 reviews the trigonometric functions, while a detailed presentation of the inverse trigonometric functions is presented in section 6.7. If desired, section 6.2 on Inverse Functions and section 6.7 may be used without modification along with the Chapter 0 material. Section 0.6 reviews the exponential and logarithmic functions, which are used extensively throughout the book. Section 0.8 can be used as a one-day bridge to calculus, illustrating the ideas that distinguish calculus from pre-calculus. Section 0.8 concludes this chapter by presenting a preview of calculus by means of the concept of limit introduced in a familiar context.
Chapter 1: Limits and Continuity - 1.1 The Concept of Limit
- 1.2 Computation of Limits
- 1.3 Continuity and its Consequences
- 1.4 Limits Involving Infinity
- 1.5 Formal Definition of the Limit
- 1.6 Limits and Loss-of-Significance Errors
Chapter 1 introduces the central concepts of limit and continuity. Limits are introduced graphically and numerically in Section 1.1, with computational rules developed in Section 1.2. Complete coverage of the formal definition of the limit is provided in Section 1.5, although this material is not required for the remainder of the text. Section 1.6 provides important insight into the computational and graphical accuracy of computers. This section may also be covered independently of other sections.
Chapter 2: Differentiation: Algebraic, Trigonometric, Exponential and Logarithmic Functions - 2.1 Tangent Lines and Velocity
- 2.2 The Derivative
- 2.3 Computation of Derivatives: The Power Rule
- 2.4 The Product and Quotient Rules
- 2.5 Derivatives of Trigonometric Functions
- 2.6 Derivatives of Exponential and Logarithmic Functions
- 2.7 The Chain Rule
- 2.8 Implicit Differentiation and Related Rates
- 2.9 The Mean Value Theorem
Chapter 2 introduces the derivative and presents the basic rules of differentiation, including the product, quotient and chain rules. The derivatives of algebraic, exponential, logarithmic and trigonometric functions are developed, providing the opportunity to present a rich set of examples of chain rules, product rules, quotient rules and applications.
Chapter 3: Applications of Differentiation - 3.1 Linear Approximations and L'Hopital's Rule
- 3.2 Newton's Method
- 3.3 Maximum and Minimum Values
- 3.4 Increasing and Decreasing Functions
- 3.5 Concavity
- 3.6 Overview of Curve Sketching
- 3.7 Optimization
- 3.8 Rates of Change in Applications
Chapter 3 presents applications of the derivative, beginning with a discussion of linear approximations and an introduction to L'Hopital's Rule. A further treatment of L'Hopital's Rule can be found in Chapter 7. Numerical methods are introduced by means of Newton's Method. A thorough development of graphical interpretations of the derivative is followed by sections on optimization and a variety of rates of change.
Chapter 4: Integration - 4.1 Antiderivatives
- 4.2 Sums and Sigma Notation
- 4.3 Area
- 4.4 The Definite Integral
- 4.5 The Fundamental Theorem of Calculus
- 4.6 Integration by Substitution
- 4.7 Numerical Integration
Chapter 4 provides an introduction to integration, starting with the process of antidifferentiation. Technology is used extensively in Chapter 4 to compute Riemann sums and observe their convergence, as well as to develop numerical integration techniques. The computation of distance from velocity provides a unifying theme to the chapter.
Chapter 5: Applications of the Definite Integral - 5.1 Area Between Curves
- 5.2 Volume
- 5.3 Volumes by Cylindrical Shells
- 5.4 Arc Length and Surface Area
- 5.5 Projectile Motion
- 5.6 Work, Moments, and Hydrostatic Force
- 5.7 Probability
Chapter 5 presents applications of integration, focusing on the development of integral formulas, routinely constructing approximating sums and then passing to the limit to obtain a definite integral. In addition to discussions of the computations of area, volume, and work, sections are included on projectile motion and also probability theory, recognizing the increasing use of statistical methods in modern society.
Chapter 6: Exponentials, Logarithms and Other Transcendental Functions - 6.1 The Natural Logarithm Revisited
- 6.2 Inverse Functions
- 6.3 The Exponential Function Revisited
- 6.4 Growth and Decay Problems
- 6.5 Separable Differential Equations
- 6.6 Euler's Method
- 6.7 The Inverse Trigonometric Functions
- 6.8 The Calculus of the Inverse Trigonometric Functions
- 6.9 The Hyperbolic Functions
Chapter 6 includes a thorough development of the exponential and logarithmic functions. Although these functions have been used throughout the preceding chapters, the complete derivation of the formulas is not provided until Chapter 6. This chapter's discussion of exponential growth and decay leads naturally into an introduction to first order differential equations. The inverse trigonometric functions and the hyperbolic functions are also discussed in Chapter 6.
Chapter 7: Integration Techniques - 7.1 Review of Formulas and Techniques
- 7.2 Integration by Parts
- 7.3 Trigonometric Techniques of Integration
- 7.4 Integration of Rational Functions using Partial Fractions
- 7.5 Integration Tables and Computer Algebra Systems
- 7.6 Indeterminate Forms and L'Hopital's Rule
- 7.7 Improper Integrals
Chapter 7 provides a variety of techniques of integration. The authors believe that students gain understanding and maturity while learning to choose among different techniques, while recognizing that computer algebra systems are routinely used to find antiderivatives. The authors include the most important techniques of integration and leave room for an instructor to cover other topics. L'Hopital's Rule is revisited in this chapter, following its introduction in Chapter 3. Chapter 7 also includes a section on the use of integral tables and computer algebra systems.
Chapter 8: Infinite Series - 8.1 Sequences of Real Numbers
- 8.2 Infinite Series
- 8.3 The Integral Test and Comparison Tests
- 8.4 Alternating Series
- 8.5 Absolute Convergence and the Ratio Test
- 8.6 Power Series
- 8.7 Taylor Series
- 8.8 Applications of Taylor Series
- 8.9 Fourier Series
Chapter 8 presents a thorough coverage of infinite series. Numerous tables of calculations and graphs are included to give students every chance to understand the difficult topic of series. Section 8.8 introduces numerous applications of Taylor series. Because Fourier series are used widely by engineers and scientists, a section on this important topic has been included in this chapter, in addition to several interesting applications.
Chapter 9: Parametric Equations and Polar - 9.1 Plane Curves and Parametric Equations
- 9.2 Calculus and Parametric Equations
- 9.3 Arc Length and Surface Area in Parametric Equations
- 9.4 Polar Coordinates
- 9.5 Calculus and Polar Coordinates
- 9.6 Conic Sections
- 9.7 Conic Sections in Polar Coordinates
Chapter 9 introduces parametric equations and polar coordinates. A large number of parametric graphs and applications are included, made more accessible by computer graphics and equation-solving capabilities. We use examples within this chapter to help the conceptual flow of material, using the amusement park ride The Scrambler as the foundation of each example.
Chapter 10: Vectors and the Geometry of Space - 10.1 Vectors in the Plane
- 10.2 Vectors in Space
- 10.3 The Dot Product
- 10.4 The Cross Product
- 10.5 Lines and Planes in Space
- 10.6 Surfaces in Space
Chapter 10 introduces students to a third dimension of graphing and calculations. Computer graphics are a valuable aid in this chapter, and are used extensively. A discussion of Magnus force relates vectors to a variety of sports applications, while providing students with practice at thinking in three-dimensional space.
Chapter 11: Vector-Valued Functions - 11.1 Vector-Valued Functions
- 11.2 The Calculus of Vector-Valued Functions
- 11.3 Motion in Space
- 11.4 Curvature
- 11.5 Tangent and Normal Vectors
Chapter 11 develops the calculus of vector-valued functions. As the graphs become more complicated, the authors' use of computer graphics increases. To keep students thinking and not simply pushing buttons, several of the examples and exercises involve matching functions and graphs, with students using the properties of functions to identify the graphs. Section 11.5 includes the important derivation of Kepler's laws.
Chapter 12: Functions of Several Variables and Differentiation - 12.1 Functions of Several Variables
- 12.2 Limits and Continuity
- 12.3 Partial Derivatives
- 12.4 Tangent Planes and Linear Approximations
- 12.5 The Chain Rule
- 12.6 The Gradient and Directional Derivatives
- 12.7 Extrema of Functions of Several Variables
- 12.8 Constrained Optimization and Lagrange Multipliers
Chapter 12 presents the calculus of functions of two or more variables. Given the increasing difficulty of visualizing the mathematics in this chapter, the Rule of Three is particularly useful in this chapter. The authors use wireframe graphs without sophisticated shading in this chapter so that students can see the traces, and not lose the details that shaded graphs tend to obscure. Where appropriate, three-dimensional graphs are augmented with contour plots and density plots. Numerically, a steepest ascent (descent) algorithm is provided, requiring some computer assistance, but reinforcing several important concepts of the calculus of functions of several variables.
Chapter 13: Multiple Integrals - 13.1 Double Integrals
- 13.2 Area, Volume and Center of Mass
- 13.3 Double Integrals in Polar Coordinates
- 13.5 Triple Integrals
- 13.6 Cylindrical Coordinates
- 13.7 Spherical Coordinates
- 13.8 Change of Variables in Multiple Integrals
Chapter 13 introduces double and triple integrals. Considerable emphasis is made on helping students develop insight into the proper coordinate systems and order of integration to use to simplify a given multiple integral. Applications involving the design of rockets and baseball bats are used to enliven the discussion of moments and centers of mass.
Chapter 14: Vector Calculus - 14.1 Vector Fields
- 14.2 Line Integrals
- 14.3 Independence of Path and Conservative Vector Fields
- 14.4 Green's Theorem
- 14.5 Curl and Divergence
- 14.6 Surface Integrals
- 14.7 The Divergence Theorem
- 14.8 Stokes' Theorem
Chapter 14 introduces the vector calculus that is essential to an understanding of fluid mechanics and applications in electricity and magnetism. Numerous graphs of vector fields are included, as well as a thorough discussion of various interpretations of these graphs.
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2002 McGraw-Hill Higher Education