The Calculus Lifesaver: All the Tools You Need to Excel at Calculus

For many students, calculus can be the most mystifying and frustrating course they will ever take. The Calculus Lifesaver provides students with the essential tools they need not only to learn calculus, but to excel at it.

All of the material in this user-friendly study guide has been proven to get results. The book arose from Adrian Banner’s popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A’s but get only average grades on exams. The complete course will be available for free on the Web in a series of videotaped lectures. This study guide works as a supplement to any single-variable calculus course or textbook. Coupled with a selection of exercises, the book can also be used as a textbook in its own right. The style is informal, non-intimidating, and even entertaining, without sacrificing comprehensiveness. The author elaborates standard course material with scores of detailed examples that treat the reader to an “inner monologue”–the train of thought students should be following in order to solve the problem–providing the necessary reasoning as well as the solution. The book’s emphasis is on building problem-solving skills. Examples range from easy to difficult and illustrate the in-depth presentation of theory.

The Calculus Lifesaver combines ease of use and readability with the depth of content and mathematical rigor of the best calculus textbooks. It is an indispensable volume for any student seeking to master calculus.

• Serves as a companion to any single-variable calculus textbook
• Informal, entertaining, and not intimidating
• Informative videos that follow the book–a full forty-eight hours of Banner’s Princeton calculus-review course–is available at Adrian Banner lectures
• More than 475 examples (ranging from easy to hard) provide step-by-step reasoning
• Theorems and methods justified and connections made to actual practice
• Difficult topics such as improper integrals and infinite series covered in detail
• Tried and tested by students taking freshman calculus

Table of Contents

Chapter 1 Functions, Graphs, and Lines
Chapter 2 Review of Trigonometry
Chapter 3 Introduction to Limits
Chapter 4 How to Solve Limit Problems Involving Polynomials
Chapter 5 Continuity and Dierentiability
Chapter 6 How to Solve Dierentiation Problems
Chapter 7 Trig Limits and Derivatives
Chapter 8 Implicit Dierentiation and Related Rates
Chapter 9 Exponentials and Logarithms
Chapter 10 Inverse Functions and Inverse Trig Functions
Chapter 11 The Derivative and Graphs
Chapter 12 Sketching Graphs
Chapter 13 Optimization and Linearization
Chapter 14 L’H^opital’s Rule and Overview of Limits
Chapter 15 Introduction to Integration
Chapter 16 De nite Integrals
Chapter 17 The Fundamental Theorems of Calculus
Chapter 18 Techniques of Integration, Part One
Chapter 19 Techniques of Integration, Part Two
Chapter 20 Improper Integrals: Basic Concepts
Chapter 21 Improper Integrals: How to Solve Problems
Chapter 22 Sequences and Series: Basic Concepts
Chapter 23 How to Solve Series Problems
Chapter 24 Taylor Polynomials, Taylor Series, and Power Series
Chapter 25 How to Solve Estimation Problems
Chapter 26 Taylor and Power Series: How to Solve Problems
Chapter 27 Parametric Equations and Polar Coordinates
Chapter 28 Complex Numbers
Chapter 29 Volumes, Arc Lengths, and Surface Areas
Chapter 30 Dierential Equations
Appendix A Limits and Proofs
Appendix B Estimating Integrals