Calculus: A Rigorous First Course
To the Student
My advice to students reading this book is very simple: Don’t believe anything you read in this book.
Perhaps I should explain further. You may be used to studying mathematics by memorizing formulas and procedures for solving different types of problems. This method can be successful in high school math, but when you get to more advanced subjects like calculus, it doesn’t work well. There are just too many different types of problems in calculus to memorize a procedure for solving each one. And for some types of calculus problems, there is no step-by-step procedure you can follow to get the answer.
An approach that works better is to grasp the concepts of calculus, so that you can understand why problems are done the way they are. With that understanding, when you come to a problem that is a little different from ones you have seen before, you can figure out how to solve it, rather than trying to apply a memorized procedure for solving it. When calculus is approached in this way, solving problems is not just a matter of calculation; it involves reasoning.
For reasoning in mathematics to be effective, it must be held to a very high standard. Our standard will be certainty: when reasoning about a problem, our goal will be not just to determine the answer, but to become certain of the answer. This pursuit of certainty should be evident throughout this book. Whenever we solve a problem, we will present a solution that is intended not merely to find the answer, but to convince you, with complete certainty, that the answer is correct. You should read this book with a skeptical attitude, refusing to believe that an answer is correct unless the solution is completely convincing. (That’s why you shouldn’t believe anything you read in this book.) And you should take a similar skeptical attitude toward your own solutions to problems: your goal is not merely to get the answer, but to be certain of the answer.
This skeptical attitude is important for success in calculus. In calculus there are often many different approaches that could be taken to a problem, some of which work and some of which don’t. It is sometimes impossible to know in advance which approach will work. When trying to solve a problem, you may have to try one approach, recognize that it doesn’t work, and then switch to a different approach. Thus, success in calculus requires not only the ability to find correct solutions, but also the ability to reject incorrect ones.
In your previous study of mathematics, you may have focused mainly on learning how to find correct solutions. Learning to recognize, and reject, incorrect solutions may be a new skill for you. How do you know when to reject a proposed solution? The best answer we can give is that you must insist on certainty. If your reasoning on a problem does not completely convince you of the answer, then it is insufficient and must be either improved or rejected. Your best defense against incorrect solutions is the skeptical attitude that we try to encourage in this book.
For reasoning to achieve certainty, it must be expressed with precision. We introduce many technical terms in this book, and when a term is introduced we always provide a precise definition. It is important to understand that terminology and notation in mathematics are always used to mean exactly what the definitions say—no more and no less. You should pay close attention to definitions, referring back to them if necessary. In many cases, the best way to get started on a problem is to be guided by the definitions of the words and notation appearing in the statement of the problem.
Often the methods we use to solve problems are based on general principles that are stated in the form of theorems. We have provided proofs of almost all of the theorems stated in this book. You may choose to skip some of these proofs, especially on a first reading. But we hope that your skeptical attitude will make you want to read them, so that you can be convinced that the theorems are true, rather than merely accepting them. These proofs demonstrate one of the fundamental principles of mathematics, without which the skeptical approach would be impossible: everything in math has a reason. Reading the proof of a theorem can not only help you understand why the theorem is true, it can also deepen your understanding of the meaning of the theorem. The proofs also provide you with good models of how mathematical reasoning should be carried out and expressed in writing.
This book requires no previous knowledge of calculus, but it does require a good background in algebra and trigonometry. Chapter 1 gives a brief review of the ideas from algebra and trigonometry that will be most important to us. If this review is not sufficient for you, then you may need to refer back to other resources on algebra and trigonometry.
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