Smith/Minton Table of Contents

Table of Contents

The vast majority of the topics found in this book are part of the standard calculus curriculum that has defined the mainstream for the last thirty years or so. The authors believe that this curriculum still has merit in terms of both mathematical precision and student learning. Nevertheless, they have made a small number of significant changes in the table of contents. After reviewing the basic properties of exponential and trigonometric functions in Chapter 0, the authors make substantial use of these functions to develop limits, derivatives and integrals. The inclusion of these functions from early in the first semester of calculus greatly increases the ability to discuss interesting applications. The authors' experience teaching in this manner has shown that this increase does not come at the expense of student understanding.

The treatment of differential equations varies widely among current calculus texts. Smith and Minton have found that many students need a sound background in integration to fully appreciate the concept of the solution of a differential equation. On the other hand, simple techniques for solving differential equations are fully accessible to second-semester calculus students and are required of many second-semester engineering students. For these reasons, the authors discuss separable differential equations in section 6.5, two chapters after the introduction of the integral and in conjunction with exponential growth and decay. In order to maximize the flexibility in using this text, they chose to introduce this material prior to Chapter 7 on techniques of integration.

Smith and Minton have not drastically revised the traditional table of contents, but rather carefully reconsidered the best way in which to present each topic. Their primary objective is to keep students focused on the central concepts of calculus. To that end, they have augmented a simple algebraic presentation of certain ideas with numerical methods. For instance, when they introduce the notion of area, they emphasize the computation of area as a limit of a Riemann sum, but use regular partitions exclusively. They do not introduce the notion of the norm of a partition until Chapter 13, when they develop multiple integrals. By that point, students should already be comfortable with the concept of the definite integral as a limit of a sum and this refinement should only enhance their understanding. They are careful to point out that (without the Fundamental Theorem of Calculus) the limit of Riemann sums can be computed directly only for a very small number of functions (polynomials of very low degree). In addition, they allow students to explore the same ideas numerically. In this case, they are not restricted to polynomials of low degree and students can observe numerical values of Riemann sums approaching a limit. With this approach, students get to see the same problem from several different viewpoints, thus improving the likelihood that they will grasp the underlying concept. Additionally, students are given a useful tool (numerical integration) which they can bring to bear on a wide variety of problems.

The authors feel that techniques of integration are of great importance. Their emphasis is on helping students develop the ability to carefully distinguish among similar-looking integrals and identify the appropriate technique of integration to apply to each integral. The attention to detail and mathematical sophistication required by this process are invaluable skills. Smith and Minton do not attempt to be encyclopedic about techniques of integration, especially given the widespread use of computer algebra systems. Section 7.5 includes a discussion of integration tables and the use of computer algebra systems for performing symbolic integration.

Flexible Topic Coverage

Smith and Minton have included a number of optional sections that are not generally found in other calculus texts, as well as expanded coverage of selected topics. These expanded and optional sections provide instructors with the flexibility to tailor their courses to the interests and abilities of each class.

In section 1.6, they explore loss-of-significance errors. Students learn how computers and calculators perform arithmetic operations and how these can cause errors, in the context of numerical approximation of limits.
In section 3.7, they present a number of diverse applications of differentiation, including chemical reaction rates and heart rates.
Direction fields and Euler's method for first order ordinary differential equations are discussed in section 6.6.
In Chapter 8, the discussion of power series and Taylor's Theorem is followed with a section on Fourier series.
Chapter 9 provides expanded coverage of parametric equations.
Section 10.4 includes a discussion of Magnus force.

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Annotated Brief Table of Contents

Chapter	Notes
0 Preliminaries	Chapter 0 is largely review, where students can reacquaint themselves with some important ideas from precalculus mathematics. The chapter also focuses students' attention on those aspects of algebra and trigonometry that are going to be most useful to them as they proceed through their study of calculus. The chapter closes with a "Preview of Calculus" section, where the concept of limit is introduced in a familiar context. .
1 Limits and Continuity	Chapter 1 introduces the central concepts of limit and continuity. The book begins with limits for a number of reasons. The most important reason is that the limit is what calculus is all about and we want students to recognize this pivotal role. The goal is for the limit-based applications presented to provide the needed practical motivation. The discussion of limits here includes limits at infinity and infinite limits. The chapter concludes with optional sections on the formal definition of limit and loss-of-significance errors in numerical computations.
2 Differentiation	In Chapter 2, the derivatives of power, exponential, logarithmic and trigonometric functions have been developed. This allows for the presentation of a rich set of examples of chain rules, product rules, quotient rules and applications.
3 Applications of Differentiation	Chapter 3 presents applications of the derivative. The first section discusses Newton's method and linear approximations. This was done because it is important for students to have a basic understanding of how computers solve equations and Newton's method is an obvious starting place. Further, the idea of linear approximation is central to many different numerical methods. Newton's method is utilized throughout the chapter as students solve for zeros and extrema of functions.
4 Integration	Chapter 4 introduces integration. For simplicity, discussions are limited to regular partitions of an interval, leaving irregular partitions for Chapter 13. Technology is utilized extensively here to compute Riemann sums and observe their convergence. It has been discovered that such computations greatly enhance students' understanding of integration as a limit.
5 Applications of the Definite Integral	Chapter 5 presents applications of integration. The focus on the development of integral formulas, routinely constructing approximating sums and then passing to the integral as a limit. Along with the standard computations of area, volume and work, sections on projectile motion and probability theory have been included. All calculus texts have a variety of projectile problems scattered throughout, but the authors felt that the topic was interesting and useful enough to warrant a more organized treatment. Probability has been labeled as an optional section, although the increasing use of statistical methods in modern society calls for a more prominent role for probability in students' education.
6 Factoring Polynomials	Chapter 6 provides a thorough and traditional development of the exponential and logarithmic functions. These functions are used throughout the text, but with frequent warnings that the derivations of formulas have been incomplete. The details are provided here. At the same time, a discussion of exponential growth and decay leads naturally into an introduction to first order differential equations. Inverse trigonometric functions and the hyperbolic functions are also introduced in this chapter.
7 Integration Techniques	Chapter 7 includes a variety of techniques of integration. The authors believe that students gain understanding and maturity while discriminating among different techniques, but they also recognize that in practice most users of the calculus routinely find antiderivatives using computer algebra systems. They try to strike a good balance of learning the most important techniques while leaving room in the syllabus for other topics. Included is a section on the use of integral tables and computer algebra systems for finding antiderivatives.
8 Infinite Series	Chapter 8 is the longest in the book, partly in response to the special difficulties many students experience when studying infinite series. Numerous tables of calculations and graphs are included to give students every chance to understand series. Since Fourier series are widely used by practitioners in engineering and the sciences, a section has been included to introduce students to this important topic and provide several interesting applications.
9 Plane Curves, Parametric Equations and Polar Coordinates	Chapter 9 introduces parametric equations and polar coordinates. A large number of parametric graphs and applications have been included, these are made more accessible by computer graphics and equation-solving capabilities.
10 Vectors and the Geometry of Space	Chapter 10 introduces a third dimension into graphing and calculations. Here again, computer graphics are a valuable aid. A discussion of the Magnus force relates vectors to a variety of sports applications, while giving students practice at thinking in three-dimensional space.
11 Vector-Valued Functions	Chapter 11 develops the calculus of vector-valued functions. The reliance on computer graphics increases as the graphs become more complicated. To keep students thinking and not simply pushing buttons, several of the examples and exercises involve matching functions and graphs, where students use the properties of functions to identify the graphs. The chapter is closed with a derivation of Kepler's laws, one of the great achievements of calculus and the human mind.
12 Functions of Several Variables and Differentiation	Chapter 12 focuses on functions of two or more variables. With the mathematics getting more difficult to visualize, it is more important than ever to follow through on the Rule of Three. Many of the computer-generated graphs are wireframe graphs without sophisticated shading. The authors have found that students can see the traces in a wireframe graph, but sometimes lose some details in a slickly produced, shaded graph. The authors also augment their three-dimensional graphs with contour plots and density plots, where appropriate. Numerically, the text presents a steepest ascent (descent) algorithm. The calculations here require computer assistance, but the algorithm nicely reinforces several important concepts of the calculus of several variables.
13 Multiple Integrals	Chapter 13 introduces double and triple integrals. The focus here is on helping students develop insight into the proper coordinate system and order of integration to use to simplify a given integral. The authors enliven the traditional topic of centers of mass and moments with calculations involving the design of rockets and baseball bats.
14 Vector Calculus	Chapter 14 introduces the vector calculus that is essential to an understanding of fluid mechanics and applications in electricity and magnetism. Reasonably simple explanations of fluid mechanics to explain and motivate the material is used. In the process, numerous graphs of vector fields are generated, and interpretations of these graphs are discussed.

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