• More than 20% of the exercises are new:
Basic exercises have been added, where appropriate, near the beginning of exercise sets. These exercises are intended to build student confidence and reinforce understanding of the fundamental concepts of a section. Some new exercises include graphs intended to encourage students to understand how a graph facilitates the solution of a problem; these exercises complement subsequent exercises in which students need to supply their own graph. Some exercises have been structured in two stages, where part (a) asks for the setup and part (b) is the evaluation. This allows students to check their answer to part (a) before completing the problem. Some challenging and extended exercises have been added toward the end of selected exercise sets. Titles have been added to selected exercises when the exercise extends a concept discussed in the section.
• New examples have been added, and additional steps have been added to the solutions of some existing examples.
• Several sections have been restructured and new subheads added to focus the organization around key concepts.
• Many new graphs and illustrations have been added, and existing ones updated, to provide additional graphical insights into key concepts.
• A few new topics have been added and others expanded (within a section or in extended exercises) that were requested by reviewers.
• New projects have been added and some existing projects have been updated.
• Alternating series and absolute convergence are now covered in one section (10.5).
• Conceptual Exercises
The most important way to foster conceptual understanding is through the problems that the instructor assigns. To that end we have included various types of problems. Some exercise sets begin with requests to explain the meanings of the basic concepts of the section and most exercise sets contain exercises designed to reinforce basic understanding. Other exercises test conceptual understanding through graphs or tables. Many exercises provide a graph to aid in visualization. Another type of exercise uses verbal descriptions to gauge conceptual understanding. We particularly value problems that combine and compare graphical, numerical, and algebraic approaches.
• Graded Exercise Sets
Each exercise set is carefully graded, progressing from basic conceptual exercises, to skill-development and graphical exercises, and then to more challenging exercises that often extend the concepts of the section, draw on concepts from previous sections, or involve applications or proofs.
• Real-World Data
Real-world data provide a tangible way to introduce, motivate, or illustrate the concepts of calculus. As a result, many of the examples and exercises deal with functions defined by such numerical data or graphs. These real-world data have been obtained by contacting companies and government agencies as well as researching on the Internet and in libraries.
One way of involving students and making them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplishment when completed. Applied Projects involve applications that are designed to appeal to the imagination of students. Discovery Projects anticipate results to be discussed later or encourage discovery through pattern recognition. Other discovery projects explore aspects of geometry: tetrahedra, hyperspheres, and intersections of three cylinders.
When using technology, it is particularly important to clearly understand the concepts that underlie the images on the screen or the results of a calculation. When properly used, graphing calculators and computers are powerful tools for discovering and understanding those concepts. This textbook can be used either with or without technology-we use two special symbols to indicate clearly when a particular type of assistance from technology is required. The icon EB indicates an exercise that definitely requires the use of graphing software or a graphing calculator to aid in sketching a graph. (That is not to say that the technology can't be used on the other exercises as well.) The symbol [!] means that the assistance of software or a graphing calculator is needed beyond just graphing to complete the exercise. Freely available websites such as WolframAlpha.com or Symbolab.com are often suitable. In cases where the full resources of a computer algebra system, such as Maple or Mathematica, are needed, we state this in the exercise. Of course, technology doesn't make pencil and paper obsolete. Hand calculation and sketches are often preferable to technology for illustrating and reinforcing some concepts. Both instructors and students need to develop the ability to decide where using technology is appropriate and where more insight is gained by working out an exercise by hand.
The late James Stewart received his M.S. from Stanford University and his Ph.D. from the University of Toronto. He conducted research at the University of London and was influenced by the famous mathematician George Polya at Stanford University. Dr. Stewart most recently served as a professor of mathematics at McMaster University, and his research focused on harmonic analysis. Dr. Stewart authored a best-selling calculus textbook series, including CALCULUS, CALCULUS: EARLY TRANSCENDENTALS and CALCULUS: CONCEPTS AND CONTEXTS as well as a series of successful precalculus texts.
1. FUNCTIONS AND LIMITS.
3. APPLICATION OF DIFFERENTIATION.
5. APPLICATIONS OF INTEGRATION.
6. INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS.
7. TECHNIQUES OF INTEGRATION.
8. FURTHER APPLICATIONS OF INTEGRATION.
9. PARAMETRIC EQUATIONS AND POLAR COORDINATES.
10. SEQUENCES, SERIES, AND POWER SERIES.
11. VECTOR AND THE GEOMETRY OF SPACE.
12. PARTIAL DERIVATIVES.
13. MULTIPLE INTEGRALS.