2007 September 6 / E-Mail Me

How does one learn a subject or study for an exam? Reread the book? Review the class notes? Make flash cards to memorize the big concepts? When studying math these are not useless approaches, but they are not the most useful, either. In my experience, the best way to study math is simply to *work as many examples and problems as possible*. This is not to say that math is about problems instead of concepts; it's just that doing problems forces you to learn about the concepts *actively*.

So how do you "work as many examples and problems as possible"? Here is a list of suggestions, from the most basic to the most ambitious. They take a lot of time and hard work, so those students who want to learn calculus well and earn a high grade will get started early. The suggestions are not unrealistic; almost every time I teach calculus, some student of mine does *all* of them (and earns an A).

These suggestions increase in potency when you study with a friend. Work together!

You simply must do *all* of the assigned problems. Do not do them just before they are due or just before the exam; do them as they are assigned, throughout the term, on a regular schedule.

Some students do *most* of the problems, but then leave off the toughest problems (which often come at the end of the section). That's not a good idea. Your exam will include some difficult problems. If you don't practice them, then you're sacrificing perhaps 25% of your exam grade right off the bat. If there are *any* assigned problems that you get wrong or cannot solve, then talk to someone about them. If you find that you cannot solve many problems at all, then talk to your professor *immediately*. In my experience, the issue is usually that the student is rusty on material from earlier math courses such as algebra, geometry, and trigonometry. Going over problems and reviewing concepts and skills is the whole point of office hours, the Math Skills Center, etc. Make use of them.

I've watched many students work out many math problems. In my opinion, they often leave off the most important step. When you get to the end of a problem, don't immediately move on to the next problem; instead, take a moment to look over your solution.

Does your solution make sense? Do you remember what your method was, or have you already forgotten it? If you were presented with the same problem again, could you solve it without hesitation? Which equations, formulas, and theorems did you use? Is there a better way to solve the problem? Maybe you did some unnecessary steps that could be avoided.

It should go without saying that when preparing for an exam you'll want to look over your old assignments and exams. For the problems you solved correctly, review your method of solution and make sure that it's sensible and efficient. If you missed any problems, then do them again until you understand them.

Remember that the assigned homework problems are the bare minimum. To improve your chances of getting a good grade, you should do many more. Simply march through the book, doing all of the problems.

Okay, you probably don't have time for *all* of the problems, so you have to exercise some discretion. Often there are a few problems of one type, then a few of another type, then a few of another type, and so on. Make sure you practice each type. Spend more time on types that you find difficult.

Many students have trouble interpreting story problems. To practice this skill specifically, march through those book sections that have many story problems (related rates, optimization, etc.); for each problem, set it up in mathematical notation, make sure the set up makes sense, and then *do not solve it* — just go on to set up the next problem. If you can't tell whether your set up makes sense, then consult with another student, with the someone in the Math Skills Center, or with your professor.

Try altering the book problems to make new problems. Many students seem intimidated by this suggestion, so I give a concrete example below, but the basic idea is this. Alter a problem and try to solve it. Repeat a few times. Then try to abstract the problem, to discover the general structure of all such problems. Or try to come up with an entirely new kind of problem that tests the same concept or skill. Or try to make a story problem out of it. Exchange problems with other students, or talk to your professor for new problem ideas.

Coming up with your own problems not only gives you extra practice, but it gets you thinking about what problems are reasonable for calculus students and how far the concepts can be pushed. Also, it puts you into the mindset of someone writing an exam; that's not useful for your life's education, but it is useful for getting a good grade.

Actually, this suggestion is just a strengthening of the "Think about how you do the problems" suggestion. Whenever you solve a problem, you should ask what makes that problem important, what its essential idea is, what other versions of the problem exist, what version might be asked on the test, what makes it different from other problems, what other kinds of problems is it related to, etc.

Suppose we've been studying differentiation and in particular the product rule. The book asks me to differentiate

*f*(*x*) = e^{x} cos(*x*) (5*x* + 1)^{4}.

Try it yourself. The function *f* is a product of three functions, so differentiating it requires two separate uses of the product rule, along with the chain rule and some careful algebra.

After solving the problem I ask, "What was essential about it?" Certainly it is about the product rule, but what makes it different from other product rule questions? I'd say it's the fact that there are three functions multiplied together, not just two. I immediately try a variation:

*f*(*x*) = e^{x} sin(*x*) (7*x*^{2} - 1)^{3}.

That's good practice, but it's pretty similar to the first problem. So next I try a wilder variation.

*f*(*x*) = arctan(*x*^{2} + 17*x*) sin(e^{2x}) (ln(*x* + 100) - 4).

These are not easy differentiation problems, and the answers are not in the book, so I ask someone to check my work.

After doing a few more variations, I get bored. I realize that I don't really care which three functions I'm talking about, but just the fact that there are three functions. So I *abstract* the problem: I ask for the general formula for the derivative of a product of three functions

*f* = *p* *q* *r*.

Following the same process as in the examples, I discover the general formula

*f*' = (*p* *q* *r*)' = *p*' *q* *r* + *p* *q*' *r* + *p* *q* *r*'.

Then maybe I wonder what the general formula for the derivative of a product of *four* functions is. What about five functions? What about 17 functions? What about *n* functions, for any positive integer *n*? Is there a simple pattern? In this way I move from a concrete book problem to a general theorem about differentiating products of many functions.

In fact, this is how a lot of math is actually discovered/invented in the first place. When you go through such a process, you are acting like a mathematician, which might be a good way to learn math.

Then I return to the original question and try to turn it into a story problem. What real-world situation requires me to multiply three things together? Well, how about finding the volume of a box from its width, length, and depth? But I need the width, length, and depth to be functions. Maybe I should let the independent variable *x* be time, and then imagine a real-world situation in which a box's width, length, and depth are changing over time. Hey, how about an ice cube melting in a glass of water?

So suppose I have an ice cube in a glass of water. At time *x* (measured in minutes, say) its width is e^{x}, its length is cos(*x*), and its depth is (5*x* + 1)^{4}. How fast is its volume changing at time *x*? I have just turned the original question into a story problem.

Is it a good story problem? I don't think so. For one thing, the width and depth are increasing over time, whereas they should probably decrease as the ice melts. So I change e^{x} to e^{-x} and (5*x* + 1)^{4} to -(5*x* + 1)^{4}. Then the width, length, and depth are all positive and decreasing at *x* = 0, which seems good.

Then I start asking myself more questions. When is the volume zero? What happens after that? Can I make the problem more realistic? Maybe the ice cube is really an ice berg and I'm observing how it melts as part of a study of climate change.

In order to design a good story problem, I had to interpret the formulas geometrically, think about their physical relevance, understand whether certain functions were positive or not and decreasing or not, etc. Making up a problem gives you a lot of practice in calculus! Then you give the problem to your friend and they get practice, too — especially if they don't know all of the steps that you took to make the problem.

As a final challenge to myself, I try to invent a story problem that's not about the volume of a box. In what other situation might I multiply three things? In this case I will use units to guide myself to a fairly reasonable problem.

Suppose I am an environmental consultant for the government of India, trying to devise a plan for the protection of tiger habitat. Depending on the number *x* of rupees I'm able to spend, I can support *p*(*x*) tigers per square kilometer of land, I can purchase *q*(*x*) square kilometers of land per year, and I can run the program for *r*(*x*) years. Then the number of tigers protected by *x* rupees is

*f*(*x*) = (*p*(*x*) tigers/km^{2}) (*q*(*x*) km^{2}/yr) (*r*(*x*) yr) = *p*(*x*) *q*(*x*) *r*(*x*) tigers.

To find how much money I should request from Indian parliament for this tiger program, I want to maximize *f*(*x*). Or do I want to maximize *f*'(*x*)? What do you think? In any event, I'm going to need to differentiate a product of three functions.

Suppose you need more practice on a particular topic, such as Newton's method, but you've already exhausted your book. In the Math Skills Center (or the library) you can find other textbooks on calculus. Pick a book and look up "Newton's method" in the index. Then march through the problems. They may be quite different from your book's problems, which is generally desirable. Your exam problems may also be different from your book problems, so it's good to practice creative problem solving. However, if the new book's problems seem too different, or if they seem to rely on a bunch of material that you've never seen before, then ask your instructor if they are appropriate for you.

The ideas of calculus are actually quite subtle and difficult. Even students who receive good grades often do not really understand what a limit is, or how one would go about proving theorems about limits. I didn't, when I first studied calculus. But if you want to improve your understanding of calculus (and your grade), you might try to understand the definitions and theorems deeply, instead of just memorizing them and doing problems with them.

Consider the Mean Value Theorem: If a function *f* is continuous on the closed interval [*a*, *b*] and differentiable on the open interval (*a*, *b*), then there is some *c* between *a* and *b* such that

*f*'(c) = (*f*(*b*) - *f*(*a*))/(*b* - *a*).

- Explain this theorem using a picture. Try to build a geometric intuition for it.
- Why are the assumptions important? What if the function were not continuous on [
*a*,*b*]? Would the conclusion still hold? Can you come up with a counterexample? What if it were not differentiable on (*a*,*b*)? - How is the theorem proved? Try to prove it yourself, but you'll probably need to consult your textbook. It will probably first prove an easier version called Rolle's Theorem. What is that? Understand how to prove the Mean Value Theorem using Rolle's Theorem. Then back up and ask how Rolle's Theorem is proved. Your book probably proves it using the Intermediate Value Theorem. What is that, and how do you prove it? If you follow the trail, then you wind up back at the definition of limits — as everything in calculus eventually does. The farther you can go along this trail, the better you'll understand the Mean Value Theorem, and much of calculus along with it.
- What is the Mean Value Theorem good for? What later theorems does it let you prove? Can you prove the Fundamental Theorem of Calculus from it? Try. Consult your book, or another calculus textbook, or your instructor.