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# Preface for Students

### Using This Textbook

If these remarks (click here) do not apply to you, then skip this section and be happy that they do not. If they do apply to you, even partially, then take what follows seriously and strive to attain the goals we state.

It would not be uncommon that you approach a mathematics textbook with the assumption that the prose in the textbook is meant to help you solve the exercises that appear at the section’s end. You see exercises as occasions to develop skilled execution of procedures and the prose as not being very necessary.

With many textbooks you would not be wrong. An unfortunate outcome of this approach is that you probably developed the habit of scanning the prose that precedes a set of exercises looking only for examples that you could mimic to solve an assigned exercise. That is, you developed the habit of attending only to the textbook’s examples, thinking that answering questions is the primary goal of your activity and that memorizing procedures is your path to this goal. The predictable outcome of this approach is that you overwhelm your short term memory with unrelated procedures for answering specific questions and you never understood the ideas that the textbook’s author hoped you would.

This textbook is different. You must not think that exercise sets are places to practice memorized procedures. The goal held by the authors of this textbook is that you understand the textbook. Exercise sets are constructed to exercise your understanding of the ideas that the textbook proposes. With an ever increasing understanding of the ideas we propose, you will find that you are able to answer complex and sophisticated questions by reasoning about them. You will answer questions successfully because you understand the ideas and relationships that the questions entail. But to reach this state you must study the text, not just look for examples to mimic.

Studying this textbook attentively will be more important than you realize. It is written to help you learn to think about important ideas. It does this by giving you opportunities to think about important ideas. But the textbook cannot think for you. To learn to think these ways you must try to think these ways. Studying the textbook attentively, striving to understand its sentences and its examples, will give you those opportunities to improve your thinking skills.

Finally, the textbook contains many animations. Do not simply watch them. Instead, strive to understand them. Pause them midway to analyze what changes, and how those changes are related to other things that change. If you cannot explain the ideas and meanings an animation illustrates, and how it illustrates them, then you do not understand it.

There are, in principle, two ways to approach a problem. One is to follow your nose, thinking only about what you have just done and what you will do next. The other is to constantly keep the original problem in mind, including what the different elements of the problem mean, and to keep in mind the reasoning you’ve done to get to this point. Keep all this in mind as you think about what to do next. To keep all these things in mind means that you must occasionally pause and mentally scan your interpretations of the problem and the reasoning you have done.

The second approach is more effortful, especially if you are unused to doing it. But the second approach will lead to significant learning if you use it consistently. The first approach might lead you to getting an answer, but you will not have learned much by getting there.

# Preface for Instructors

### Special nature of the approach (expand each into more elaborate prose).

• Calculus addresses two foundational issues:
• You know how fast a quantity changes at every moment; you want to know how much of it there is at every moment.
• You know how much of a quantity there is at every moment; you want to know how fast it is changing at every moment.
• Rooted in research on students’ mathematical thinking and understanding
• Always refer to values of variables
• Variables vary
• Differentials are at the heart of this approach. A differential is a variable that varies through (mainly small) intervals and we speak of differentials only in the context of constant rate of change. The differential dx is a varying change in x. The symbol $\Delta x$ represents the size of intervals through which dx varies. The differential dy is always a constant multiple of dx.
• We avoid “slope”. Rather we emphasize rate of change.
• The special meaning of “at a moment that $x=x_0$" is developed in Chapter 3 and is used throughout the book. By a moment of a variable we mean a small interval containing a particular value of the variable.
• We develop the concept of a function that gives the rate of change of another function at a moment of the other function’s independent variable. This is the meaning of derivative that we develop.
• An integral is not an area. It is an accumulation. If the quantity that is accumulating is area, then an integral is an area. But most times it is not.
• Antiderivatives are functions. They are not merely symbolic expressions that you differentiate.
• We do not develop derivatives, then integrals, then bring them together with the FTC. Rather, like Newton, we begin with the FTC in that rate of change and accumulation are two sides of a coin. We use the concept of rate of change to develop accumulation functions and we use the concept of accumulation to develop rate of change functions.
• We state the FTC even before developing rate of change from accumulation (derivatives). In fact, we use the FTC as motivation for finding derivatives.
• Uses of the textbook as a visual aid during class.
• Pacing: Advice on how to select review material.
• (And more when I have time to finish this.)

### Introduction

Arithmetic is about static quantities whose values you know. Algebra is about two things: static quantities whose values you do not know and relationships between quantities whose values vary.

Calculus is about changes in quantities’ values. Indeed, calculus addresses two foundational problems:

1. You know how fast a quantity is changing at every moment; you want to know how much of it there is at every moment.
2. You know how much of a quantity there is at every moment; you want to know how fast it is changing at every moment.

(Expand, explain, and justify these two statements in historical terms and in non-technical language.)

Joe Wagner …

Rob Ely …