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You are taking a course in calculus. If you are a typical student, you expect to become skilled at using procedures to answer questions at the end of each chapter.
If you have taken calculus before, you probably know about derivatives and integrals and expect to become highly skilled at differentiating and integrating functions the textbook presents to you.
However, if you hope to enter a field requiring calculus, then you should know the reality of what you face.
Any company that might hire you already has a $350 computer program or handheld app that can differentiate and integrate functions faster and more accurately than you ever will!
In other words, if all you can do is procedural calculus, you will lose to this computer program.
On the other hand:
Each of the above requires a human who understands how to conceptualize situations and create mathematical models of them. They require a human who understands the calculus well enough to justify and explain the solution he or she creates using this computer program.
These are our aims:
Even if you do not plan on a career that requires calculus, you will benefit by being literate in one of the greatest intellectual achievements of modern humanity.
But you must be patient about learning the ideas of calculus. They are very deep, and they come from understanding many elementary concepts in ways new to you. Again, be patient.
Read Michael Larner's description of a common disconnect between students and professors. If Larner's remarks do not apply to you, then skip this section and be happy they do not.
If these remarks 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 a particular attitude. It is with the assumption that the prose in it is meant to help you solve exercises appearing at the section’s end. You see exercises as occasions to develop skilled execution of procedures and the prose as being unnecessary.
With many textbooks you would be right. An unfortunate outcome of this approach is you developed the habit of scanning the textbook for examples to mimic in solving 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 memorizing procedures is your path to this goal.
The outcome of this approach is predictable. Students overwhelm their short-term memory with unrelated procedures for answering questions. Students fail to understand the ideas the textbook’s authors hoped they would.
This textbook is different. You must not think 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 in this textbook are constructed to exercise your understanding of the ideas the textbook proposes.
With an ever-increasing understanding of the ideas we propose, you will find 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 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.
Important. 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. Take this as a sign that you must speak with your instructor or TA.
There are, in principle, two ways to approach a problem.
The second approach requires more effort, 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.
Finally, do not hesitate to use your instructor's or TAs' office hours! Avoiding office hours can lead to a terminal disease.