This essay on calculus was written by Prof. Ron Freiwald of Washington University in St. Louis. Prof. Freiwald is a former teacher and, more recently, colleague of mine. I am deeply indebted to him for all of the knowledge he has imparted to me over the years and I thank him here for allowing me to share his thoughts with you. Despite being written for a different course at a different school, his words ring true.

The seeds for the main ideas of calculus go back to ancient times, but calculus itself, as we think of it today, was invented (or discovered?) during the 17th century in response to the needs of an emerging science.  Its invention is usually attributed to the English mathematician Sir Isaac Newton and the German mathematician Gottfried Leibniz, working independently. As its power developed, calculus gave scientists a tool to generate remarkable new understandings of the world. Its creation is considered one of the great intellectual accomplishments of the human mind. This sweeping assertion is justified not only by the beauty of the subject, but also by the fact that it still retains its fundamental importance, even several centuries after its birth.  In fact, its role has become more important than ever as the use of mathematical models has reached beyond areas such as physics and engineering and into such fields as biology, economics and business.

Of course there are also other important mathematical tools.  Algebra, discrete mathematics, probability and statistics, topology, and computer science all have roles to play, and these tools complement each other. The increasing power and availability of technology enhances their usefulness and doesn't replace the need for any of them. See the syllabus section Technological Tools to read more about how we will employ these tools.

There are lots of details and techniques for us to learn, but in the big picture there are only two "great  ideas" in calculus.  Both of them are illustrated on the dashboard of your car.

1)      The first idea concerns "how fast is a quantity changing?"  For example, if you're driving down the highway and s represents the distance you've traveled from home, then you might be interested in how fast
s is changing (measured, perhaps, in km/hr).  How fast s changes over time is your velocity, v.  Studying rates of change involves a concept from calculus called the derivative. It's analogous to the speedometer on your car.  Of course, instead of how fast s is changing, you might be interested in the
rate of change of

y = the size of a population of bacteria

y = the amount (mass) of a radioactive isotope in a sample

y = the price of a gallon of gasoline.

Whether a rate of change occurs in biology, physics, or economics, the same mathematical concept--the derivative--is involved in each case.
 

2)     The other idea is the "opposite" of the first one.  If you know how fast some quantity is growing, then how big is it at a certain time?  On the highway again, you could imagine trying to figure out how far s you are from home at time t by studying the velocity v (speedometer information).

This is easy to do if the driver goes at constant velocity  (in that case, s = vt ).  But finding s from v is harder if the velocity is always changing during the trip. The calculation of a total amount s from a rate v involves a concept from calculus called the integral.   It's analogous to the tripmeter, which (if you set it to 0 when you start out) measures s on your dashboard.

Instead of a distance, you might be interested in trying to compute the total amount of money in an account if you know how fast it's growing (the interest rate), or the total number of people infected with some disease at time t if you know the infection rate.  The same mathematical concept, the integral, is involved in each case.

Nearly all of Math 141-142 consists of
 
• developing the background for these ideas
• developing the mathematical concepts of derivative and integral
• seeing how the two ideas are connected
• learning techniques for applying the ideas efficiently.


All of this is hard work.  But what did you expect in learning about "one of the great intellectual accomplishments of the human mind"?  The day to day work may seem tedious at times, but it's essential, like finger exercises for the future pianist. Or, to change the analogy, it's like learning a new language: it can open new vistas but only if you're willing to memorize vocabulary, learn to conjugate verbs, and practice, practice, practice!

We'll see indications of some of the diverse applications of calculus during the course. But it's not a course in physics, biology, economics, or business.  Many of the most interesting and significant applications you will have to meet elsewhere.  That should be a relief!  It's certainly nice to get ideas about what the material is good for, but students who want "more applications" in math courses often don't realize that applications, generally, are much harder: a little like "story problems", only worse. That's because applying math to a concrete situation involves taking a complicated, messy real-life situation, deciding what's relevant to the problem and what isn't, creating a mathematical approximation ("model") to reality, and setting up a mathematical formulation of the problem. Only at that point are you ready to apply the "tools" from calculus. Setting up a mathematical model in an application is often not easy, and it sometimes requires detailed knowledge of another subject, such as physics or economics.  In a calculus course you learn the tools and see them applied in some "tidy" applications which only hint at the real usefulness of the subject. The biologists, chemists, physicists, engineers, architects, economists, and others who have recommended that you take a calculus course will have to show you the reasons why it's useful in their own fields (please, put them on the spot and ask!! )  For now, try to learn to appreciate the subject itself, its beauty, and how the pieces fit together.
 
 



Expectations and Advice

You'll probably find that the material in most university courses, including this one, is covered much more quickly than it would be in high school.  You'll probably also be asked to have a greater command of the material than before, especially in understanding the ideas (not just the techniques) and applying them in new situations. This may take some adjustments in your style of learning.

The instructors are here to help you understand and learn the material, but actually learning it is your responsibility. The lectures are designed to try to highlight the important ideas and give you some perspective on the material, so that you can digest and really learn the material as you think, read the text, discuss the material with other students, and practice on problems. You should expect that more of the learning will occur outside the classroom than in it.  On the practical side, the average student should expect to spend at least a couple of hours on calculus for every hour in the classroom.  If you do this for all your courses, then being a student is the equivalent of a full-time job.

Read the text. Your instructor can't possibly say everything that's there. An hour's lecture wouldn't be enough even to read the day's assignment aloud, let alone try to highlight or clarify. Your instructor's job is to complement the text so that you can learn from it.  Simply relying on your lecture notes will (and should!!!) probably only give you the highlights of the material.  To ignore the text is to start out looking for a minimal understanding.

Reading a math text requires active involvement and is often a slow process. You might spend a few hours reading an assignment of 100 pages in another subject.  A reading assignment of 10 pages of mathematics might take just as much time.  Read with scratch paper at hand and fill in missing steps. When the author simply asserts  "...this expression simplifies to...", simplify it!  When you read "you should check that...", check it!  Try to work some of the examples in the text before reading them.  Play with the material: ask yourself  "what if I changed the problem to...?".  Write down questions that come to mind.  If you can't answer them, ask your friends. Try to understand why the assertions in the text are true.
 

One of the marvelous features of mathematics is that you need not (perhaps should not!) trust the author.  If a physics book refers to an experimental result, it might be difficult or prohibitively expensive for you to do the experiment yourself.  If a history book describes some events, it might be highly impractical to find the original sources (which may be in a language you do not understand).  But with mathematics, all is before you to verify.  Have a reasonable attitude of doubt as you read; demand of yourself to verify the material presented.  Mathematics is not so much about the truths it espouses but about how those truths are established.  Be an active participant in the process.
                                        E.R., Scheinerman, Mathematics, A Discrete Introduction
                                        Brooks/Cole, Pacific Grove, CA, 2000, p. xviii-xix.


Ideally, you should read ahead.  The lectures will be much more valuable if you have tried to digest some of the material already.  Also, you'll be able to judge that some things on the board needn't be written down.  For example, you'll know that those formulas the instructor is writing on the board are just the ones listed in the text.  That frees you to think about what's being said and not be always frantically transcribing material from the board.

If you choose to ignore the advice about reading ahead, then at least read the material in the text before trying to do the assigned problems.  Some students start out with the problems and when they get stuck, they page back through the text looking for a formula or example to help.  This puts the focus in the wrong place.  Simply being able to "do the problems" represents at best a minimal strategy for the course. You do have to do problems, and you do want to have certain technical skills that the problems sets can help develop, but you also want to understand the material.  Carefully reading the text is essential for that.

You can't learn to speak French by merely sitting and listening, and you can't learn mathematics by watching someone else do problems. You need to do as many problems as you can realistically find time to do: the more, the better.  It's useful to work with other people on homework as long as you're involved as an active partner. Compare solutions; ask each other questions; make up quizzes for each other; agree on some extra problems to try together--such as the review problems at the end of each chapter.  Prod each other to do the assignments!  Do whatever helps!  But all of this is to improve your understanding: your study group isn't there at quizzes, tests, or when you need your math in another course.  See the syllabus section Homework Policy and Suggestions.

There are facts and formulas you need to have at your fingertips: that is, there are some things you simply need to memorize.  But that's only the beginning.  Formulas are just tools.  Some routine problems, admittedly, are designed merely to be sure you really can handle the tools, but when attacking a non-routine problem, don't just try to hunt
around for a formula. To quote an old but accurate text,
 

"(the) time wasters are the formula worshipers ... who spend more time hunting a magic formula than they would need to analyze the problem piece by piece using simple familiar methods and calculations ... it cannot be said too often that the ability to understand and solve problems does not come by memorizing formulas ... formulas are not substitutes for thought."


Many problems on tests will be "like" problems you've seen before.  But some won't be.  This isn't grounds for complaining that "we never saw a problem like that before."   If you can only do problems like ones you've seen before, what's the point?  The purpose of the problems is to help you understand the ideas and techniques, not merely to learn how to do more similar problems.

Stay on top.  Sometimes it's hard to keep up with everything, but it's important not to fall behind in the course.  A lot of the material we cover is interdependent, and if you're not comfortable with the material from two days ago, today's lecture might be totally incomprehensible.  It's better to be a bit ahead--the lectures will mean more and you've got a buffer when an emergency strikes.

Also, there's a limit to how fast you can digest material and learn actually to use it in your thinking.  Avoid cramming just before tests by staying on top of things.  It makes life much more pleasant.

Make use of your instructor.  Office hours are set aside specifically to see students; those times are for you!  Most instructors are also willing to set up special meetings with students as time permits. Your responsibility is not to waste these times.  Try to solve your own problems first, together with your friends. What you learn that way will be more valuable than the same answer "given" by your instructor. Come to office hours or your discussion section prepared with specific questions.  Bring questions to your instructor quickly, as soon as you realize there's some difficulty you can't resolve.  (It's hard for any instructor to cope with  "I haven't understood anything for three weeks"; we aren't magicians.)

Finally, one last word.  Some of you have had some calculus already: this can be a blessing or a curse.  It's a curse if it makes you think you know more than you do and traps you into slacking off until you suddenly realize you're in trouble. Even if you do already understand what's being covered in class and the homework, this shouldn't make you think there's nothing to learn.  Try instead some of the harder problems in the text so that the time's not wasted.  If you're bored and want to talk about some harder problems or find some extra reading, talk to your instructor.  If you're serious, your instructor will probably be delighted!  And, if you're on top of the current material, then you have a great opportunity to be helpful to some of your friends who are still working on it!