Bridging Traditional and Reform
Within a framework of fairly traditional topics, the books incorporate important features of the various calculus reform projects,
- Thorough integration of graphing, spreadsheet, and on-line technologies.
- Focus on applications using real data
- Emphasis on
mathematical concepts through the extensive use of conceptual exercises
and pedagogical techniques such as the Rule of Four (numerical,
geometric, algebraic, and communication-oriented approaches to
- Through our Internet site, we now add a Fifth element:
interactive discourse. The student can now go on-line and take a
quiz, play a game tutorial or interact with more helpful regular tutorial, play a zero-sum game against the computer, or watch a visual simulation of a Markov process or a limit. The possibilities are
endless, and the site continues to grow and evolve.
- Inclusion of almost all the topics found in more traditional texts.
- While the books are technology-oriented, the organization of the material has been planned to ensure that students equipped with
nothing more than a scientific calculator will not find themselves at a
- The texts are carefully structured and tightly organized for easy navigation and reference, and we have taken pains to be mathematically precise in all our definitions and statements of results.
- Abundance of practice and drill exercises
- Large numbers of application exercises to choose from
Options in Technology
The use of graphing calculators, spreadsheets, and the large collection of online utilities available on this Web site has been thoroughly integrated throughout the discussion, examples, and exercise sets, beginning with the first example of the graph of an equation in Chapter 1.
- Flexibility in Choice of Technology
We incorporate all of the following technologies: graphing calculator (based on the TI-83/84), computer spreadhseet, and more than 20 online utilities offered on this Web site. As a result, the text can be used in a classroom devoted to a single technology mode (for instance, graphing calculators only) or in a setting where instructors and students can choose different technologies for different topics.
- Margin Technology Notes
We include margin technology notes to support almost all the examples that use technology. These notes provide quick outlines of the more detailed step-by-step instructions in the end-of-chapter technology guides, and so are ideal references for students already accustomed to the use of technology.
- End-of-Chapter Technology Guides
Each Chapter ends with detailed Technology Guides for the TI-83/84 and for spreadsheets that walk the student step-by-step through all the technology-based examples discussed in the chapter.
- Influence of Technology on Material
The focus on technology plays an important conceptual and pedagogical role in our presentation of many topics. For example, our discussion of the operations of arithmetic in Chapter 0 includes a careful discussion of formula syntax for technology. Our discussion of mathematics of finance includes descriptions of using technology to solve problems normally requiring techniques not normally covered in finite mathematics courses. Our treatment of curve sketching was written with graphing technology in mind, and we have used a flexible approach that can be adapted to the increasingly popular practice of using graphing technology to draw the graphs and then using calculus to explain the results. (On the other hand, instructors who prefer "by-hand" sketching can simply ignore technology and use the text in a more traditional manner.) Some of the real power of technology is seen in the chapter on applications of the integral, where we guide the student in the use of technology to analyze mathematical models based on real data, make projections, and calculate and graph moving averages.
Our comprehensive exercise sets continue to be among the strongest features of our textbooks, and the 6th edition is no exception. Our comprehensive collection of exercises provides a wealth of material that can be used to challenge students at almost every level of preparation, and includes everything from straightforward drill exercises to interesting and rather challenging applications. We have therefore included, in virtually every section of every chapter:
- Applications Based on Real Data
A most striking distinguishing feature of these texts is the diversity, breadth and abundance of examples and exercises based on real and referenced data from business, economics, the life sciences and the social sciences. This focus on real data has contributed to the creation of a book that students in diverse fields can relate to, and that instructors can use to demonstrate the importance and relevance of calculus in the real world.
- Further expanded in 6e: Communication and Reasoning Exercises
These are designed to broaden the student's grasp of the mathematical concepts, and include exercises in which the student is asked to provide his or her own examples to illustrate a point, to design an application with a given solution, or to spot the error in a fictitious student calculation, as well as "fill in the blank" type exercises and exercises that invite discussion and debate and have no single correct answer.
- Technology Exercises
Our technology exercises have been designed for all three types of technology discussed in the books: graphing calculator, spreadsheet, and the Website technology tools, often in relation to real data where by-hand computation would be difficult or tedious.
- Revisited Themes
Many of the scenarios used in application examples and exercises will be revisited several times throughout the book. Thus, for instance, students will find themselves using a variety of techniques, from graphing through the use of derivatives to elasticity of demand, to maximize revenue in the same application.
- Graduated Difficulty Level and Exercise Hints
Exercises that are somewhat more advanced, not based wholly on examples, or sometimes require a student to think "outside the box" are designated as "more advanced" (marked with orange triangles) while exercises that are more difficult are designated as "challenging" (marked with black diamonds; click on the graphic to the right to see examples of all three types of exercise.) Hints are often included that relate exercises to specific examples.
- Further Expanded in 6e: Chapter Review Exercise Sets
The chapter review exercise sets have been expanded with the addition of many more basic skills exercises and applications. All the applications in the chapter review exercises revolve around the various business and other exploits of fictitious online seller, OHaganBooks.com and CEO John O'Hagan. The diligent reader will be able to track the college career of John O'Hagan's son Billy-Sean, puzzle through the various corporate spy scenarios in the game theory chapter, and also speculate about John O'Hagan's sometimes dubious business decisions.
Scattered throughout all the exercise sets you can find some scenarios that are tongue-in-cheek references to current events or, on occasion, just plain absurd. We hope that these will elicit a chuckle or two.
Using the Website:
Options for Teaching, Assessment, and Distance Learning
For the instructor, the resources on this Website can be combined with the book to accomodate a number of methods of instruction and student assessment. For the student, the site provides an unequalled collection or resources for individual study and review for tests.
- Distance Learning and Hybrid
The section-by-section teaching videos (there are links to the videos in the "Everything" pages) and online section-by-section tutorials can be used together as complete online pedagogy content in a distance learning class. Instructors could simply point the student to the growing collection of videos and tutorials without the need to create an extensive collection of online pedagogy typcially demanded in such courses. The on-line utilities then give the distance learning student a variety of tools for task completion without the need to use a graphing calculator. For asessment of distance learning students, the online testing and homework resources provided by both WebAssign and MindTap are available for these books, enhanced through additional videos and links to the tutorials. Further assessment of student performance can be done through the use of the "game tutorials" (tutorials with randomized questions that that work as games and give assessments of the student's performance at the end of the game) and a growing collection of online chapter review exercise sets that can be set to "test mode" to randomize questions and assess performance.
- Classroom Instruction via Online Tutorials
The on-line section-by-section tutorials cover a large and expanding number of topics in the books, and provide a convenient medium for in-class instruction. Through the numerous interactive features built in to the tutorials, along with the on-line utilities, students can participate actively in the classroom rather than passively as note-takers. The tutorials are designed to outline the main features within particular sections, preparing the student for a more in-depth reading of the textbook.
- Classroom Instruction with Online Technology Tools
To support the use of technology in the classroom, we offer a comprehensive array of on-line utilities: graphing and function evaulation utilities, regression and finance tools, matrix algebra and matrix pivoting tools, statistics utilities and graphers, and specialized utilities for tasks such as linear programming, numerical integration, Markov processes, and game theory. As indicated in the text, these utilities are ideal for classroom demonstration of everything from graphs and tables to applications of linear programming and interactive displays of Riemann, trapezoid and Simpson sums in integration.
- In-Class Student Assessment Using Game tutorials and Chapter Review exercises
The "game tutorials" are challenging tutorials with randomized questions that that work as games (complete with "health" scores, "health vials" and an assessment of one's performance at the end of the game) and are offered alongside the traditional tutorials. These game tutorials, which mirror the traditional "more gentle" tutorials, randomize all the questions and do not give the student answers but instead offer hints in exchange for "health points," so that just staying alive (not running out of health) can be quite challenging.
- As a home study and review medium
In addition to the tutorials, the student can use our detailed chapter summaries which serve as a supplementary "mini-text" with interactive elements and numerous links to related resources, online chapter review exercises that supplement those on the textbook and also provide, for a growing numer of chapters, the ability to generate randomized questions and take practice tests. In addition, the student can take quick randomized chapter true-false quizzes to test conceptual understanding of the material.
- Spanish A parallel Spanish version of the entire Web site is gradually being developed. All of the Chapter summaries and many of the tutorials, game tutorials, and utilities are already available in Spanish.
To learn more about the Web site, click here.
We would like students to read this book. We would like students to enjoy reading this book. Thus, we have written the book in a conversational and student-oriented style to encourage the development of the student's mathematical curiosity and intuition. Some unique features of our pedagogy include:
- Question-and-Answer Dialogue
We frequently use informal question-and-answer dialogues that anticipate the kind of questions that may occur to the student and also guide the student through the development of new concepts.
- Quick Examples
Most definition boxes include one or more straightforward examples that a student can use to solidify each new concept as soon as it is encountered.
These are collections of "frequently asked questions" and answers at the end of many sections whose purpose it is to answer common student questions and reinforce new concepts (Click on picture opposite to see a sample.)
- Before We Go On
Most examples are followed by supplementary interpretive discussions under the heading "Before we go on." These discussions may include a check on the answer, a discussion of the feasibility and significance of a solution, or an in-depth look at what the solution means.
- Communication and Reasoning Exercises
These are exercises designed to broaden the student's grasp of the mathematical concepts. They include exercises in which the student is asked to provide his or her own examples to illustrate a point or design an application with a given solution. They also include "fill in the blank" type exercises and exercises that invite discussion and debate. These exercises often have no single correct answer.
- Unique Pedagogical Devices
As instructors, we have all seen students encounter conceptual barriers in finite mathematics and calculus classes. We list a few we are sure you have encountered, and outline how we deal with them:
- Word Problems: Students unable to translate stements into mathematical equaitons
In the chapters on systems of linear equations and linear programming, we carefully coach the student to reword each statement in a specified way that translates easily into symbols
- Differentiation Tequniques: Students unable to decide which rule to use where
In the chapter on techniques of differentiation we describe a "thought experiment" to lead the student to a valid hierarchy of derivatives rules.
- Counting arguments: Students unsure of how to organize information
In the chapter on sets and counting we discuss "decision algorithms:" the student is urged to pretend he or she was going through all the steps in constructing, say, a poker hand of a specified type. The resulting sequence of decisions is then translated into a counting algorithm quite mechanically.
- Matrix Row Operations: Students going around in circles getting nowhere in reducing a matrix
Our dicussion of setting up row operations describes a detailed, step-by-step procedure for students to follow in setting up the appropriate row operations and ways to check the status of the computation.
Every chapter begins with the statement of an interesting problem scenario that is returned to at the end of that chapter in a section titled "Case Study." This extended application uses and illustrates the central ideas of the chapter, and can be used as a reading project, group project, or take-home test. The themes of these applications are varied, and they are designed to be as non-intimidating as possible. Thus, for example, the authors avoid pulling complicated formulas out of thin air, but focus instead on the development of mathematical models appropriate to the topics.
Some examples of Case Studies are the solution of the "diet problem" using linear programming, an analysis of adjustable rate and subprime mortgages, the use of marginal analysis to design a strategy for regulating sulfur emissions, and the use of Benford's Law to spot fraudulent tax returns. These applications are ideal for assignment as individual or group projects, and it is to this end that we have included groups of exercises at the end of each.