Preface for the Student

For many, the study of calculus is seen as a rite of passage — to conquer calculus is to pass through the gateway to the sciences, engineering, mathematics, business, economics, technology, and many other fields. For some, the study of calculus is indicative of achievement, a hallmark of a quality education. A few can’t wait to study calculus, their curiosity overflowing with enthusiasm. Yet others see calculus as an annoyance, something to tolerate in pursuit of more important or more interesting subjects. This book is for all of you.

Whatever your reason for studying calculus, it is the author ’s hope that this text facilitates not just the mastering of technical skills and the understanding of mathematical concepts, but also training in thinking in a patient, systematic, disciplined, and logical manner. While technical skills can be useful for some students in their careers, and the understanding of mathematical concepts can be of use to even more, the habits of mind created by careful thinking can be of use to everyone, at any time, in any place.

Preparation for success

If you have learned to drive a car, then you may recall how driving took much conscious thought at first, but later, with practice, driving became much more of a background task. The same is true of addition and multiplication facts; the task 2 + 4 takes very little mental energy. That is the hallmark of deep learning — when a task has been thoroughly learned then it can be performed accurately with little effort.

Success in calculus is much easier if basic algebraic and trigonometric skills have been learned that deeply. If the quadratic formula and laws of exponents can be applied accurately upon demand, then the mind is free to concentrate on the concepts at hand. If not, then instead of consciously juggling three new concepts, a dozen or more distracting items that must be re-learned compete with the new concepts for mental energy, hampering one’s learning of the new

material.

Mental skill-building is much the same as physical skill-building; it takes time and consistent effort on the part of the learner. Lifting weights several times per week for a month is a much more effective strategy than waiting until the night before the skills test to try to cram the entire month’s reps into one evening’s workout. Bodybuilding simply does not work that way, and neither does learning mathematics.

One final bit of advice: learn from failure. Everyone makes errors, even textbook authors with decades of experience in the subject. No one is perfect. But when you make an error make sure that you understand why that was an error, why a different approach must be used, and how to avoid making the same error in the future. There is often something to be learned from your errors. Mistakes are not to be feared, but to be used to your advantage!

Features of this textbook

What makes this textbook different? The most obvious answer is that it uses infinitesimals, which are infinitely small numbers that you might not have encountered in your previous courses.

Although infinitesimals were an essential part of the development of calculus, they have been absent from nearly all calculus textbooks for over a century. The largest factor in the switch away from infinitesimals was the fact that, at the time, no one had been able to rigorously develop the required number system. That fact changed in the 1960s, and now the use of infinitesimals is once again seen as mathematically legitimate. Using notation and procedures developed and published by the author, the study of certain portions of calculus in this text is both more intuitive and algebraically simpler than in other calculus textbooks.

Additional features include:

• A readable and student-friendly narrative. The narrative is written to help students think through the development of concepts and think through solutions to examples. Following the thinking process helps the student create meaning and retain the ideas more easily.

• READING EXERCISE 1

Reading exercises are meant to be worked when encountered during reading. The solution is placed in the margin one to three pages later, with the section and reading exercise number in the same yellow-orange color.

• Color-coded boxes. Theorems are in blue boxes, definitions are in green boxes, and formulas to be learned (memorized) are in brown boxes.

• Hundreds of diagrams. Consistency of color usage throughout the text’s diagrams help with interpretation.

• EXAMPLE 1

Hundreds of examples with complete solutions are included.

Some solutions are written compactly demonstrating the level of detail expected of student work. Others include more details of how to think through the solution.

• Thousands of exercises. Exercises range from the routine to the challenging. Many sections include “rapid response” exercises meant to help the student distinguish between objects or algebraic forms. Some exercises are very similar to examples in the narrative. Other exercises require the student to think creatively or explore the ideas more deeply. Sometimes exercises from much older textbooks are included, such as those labeled (GSL) from the classic early-twentieth-century text of Granville, Smith and

Longley.

• Answers to odd-numbered exercises. The answers to odd-numbered exercises sometimes include hints, brief explanations of why some attempted answers are incorrect, alternate forms of answers, or both simplified and non-simplified answers in order to help students determine the source of an error.

• An extensive index.

It is my prayer that this textbook will be a blessing to you, that it will help you understand the concepts and develop the skills of calculus as you continue your educational journey. Enjoy!

Bryan Dawson University Professor of Mathematics Union University July 2019

Margin notes. Margin notes are used to add explanations, tips, cautions against making common errors, and historical notes.

Chapter

Review