Complex Calculus Problem Solving in Seven Steps

By Terence J. Grant, August 13, 2004

Can you solve infinitely complex math problems? Problem solving is an important skill to have; in fact, everyone can problem solve to some degree. However, if you are unable to solve a problem, you might find that it can get complex, and that this can happen fairly quickly. You also might be in for a shock when you take your first “higher level” math course, and find how simple looking problems become exponentially complex.

A simple plan of attack, which often works for everyday math problems, is the “guess and check” method. In this method one simply looks at the criteria of a problem, guesses at the most likely solution method, implements it, and at the end, checks the answer to see if the desired result is obtained. The more problems that fit a certain criteria, the more likely you'll be to use the same methods of which to solve it.

Herein lies a trap; while getting set in your “guess and check” ways works great for mechanical mathematics like Algebra and Geometry, more complex mathematics like Calculus require more planning.

What happens to a lot of people in Calculus classes is that they will initially refer to answer guides, look over them, agree that the steps involved look like logical progressions, and then state something akin to, “okay, but how did they know to do that?” The answer to this is simply “practice,” however, what exactly to practice is not really known or stated.

Calculus is more concerned with “thought,” that is to say, the idea behind a problem, rather than the “mechanics,” which consists of simple number crunching, as in simpler math. The mechanics of Calculus still exist, but are typically left to the Algebra involved. However, don't make the mistake into thinking that once you solve the Calculus portions of a problem that you're just left with Algebra. While this may be true for beginning Calculus problems, what you'll find is that the more complex problems will be a mix of thought and mechanics at once. So our overall method of solving a Calculus problem is, not so surprisingly, a mixture of thought and mechanics together as well.

I have developed seven steps to which I follow for each Calculus problem I come across. I have found that since I developed this method via independent study for a Calculus II class, that this has worked for each and every problem attempted without fail. Although the techniques listed deal very well with complex Calculus problems, they are not limited to Calculus, or even math. They work well for other disciplines, and can be used to solve complex engineering problems as well as other complex problems in everyday life. Here is the list of steps in sequential order.

1. State the problem, completely.
2. Use identities to transform the problem.
3. Break the problem into pieces.
4. For each piece, decide on a plan of attack.
5. State the plan, referencing some fact.
6. Apply the fact, and if the fact proceeds to solve the piece, then decide a plan of attack on the next piece.
7. Make a general statement about the solution to the problem.

State the problem, completely. This means just what it says. If you are given an integral and asked to integrate it, write the integral along with question. This gives you a well set goal, and lets you make sure that you solve a problem completely, not just part of it.

Use identities to transform the problem. When you think of identities, think of Trigonometry, and the various ways in which you can write Sines and Cosines. Furthermore, there are also many ways to state any given problem, and some work better than others. Also, in Calculus you'll find that sometimes one given form of a problem fits a theorem if the problem is restated in another form. State all the identities you can for the problem, preferably on the same line that you've stated the original problem if you can. At this point I will also state that I believe Trigonometric identities are best stated in terms of Sine and Cosine only, when possible. Using identities to your advantage is paramount.

Break the problem into pieces. Small pieces. This goes along with using identities, however, sometimes simply stating a limit or an integral as the sum or product of two limits or two integrals (respectively) can make a problem that much easier.

For each piece, decide on a plan of attack. This is where the “mechanics” all but stop and the “thought” comes back in. Now is the time when you start researching your theorems in a more hardcore fashion. You should also look for theorems that state something in a form that is close to what you have, although this is something that will come with practice.

State the plan, referencing some fact. By “some fact” I mean a theorem or a suggested step referenced in your materials. Now this is also the part where you need to actually write down your plan of action. If you do not write down your plan, you're more than likely not going to follow it to the letter, and you'll find yourself back in “guess and check” world in no time flat. So write it down.

Apply the fact, and if the fact proceeds to solve the piece, then decide a plan of attack on the next piece. Easily stated and lived- if the piece is solved, you're well on your way to the next piece. However, if you find a plan of action didn't work, rewind to the first step and make sure you haven't made a mistake at each step down the line. Approaching a problem again from the first step often works well too, and although time consuming, practice will indeed allow you to approach perfection. Once you're confident, proceed to the next piece, or, if you're at the end, the next step.

Make a general statement about the solution to the problem. This is probably the point at which you can make your deadliest mistakes, if you're not careful. Write your answer, and make a statement about it. Typically if you're following a set of steps that requires you to write several equations, make sure you're stating the solution to the original equation, not to one of the other equations contained within.

Congratulations, you can now solve complex Calculus problems. Your answers are no longer just sets of numbers, but they contain explanations that look curiously similar to those you might find in answer books. This helps not only in understanding the material, but advancement in the subject as well. There's no doubt in my mind that anyone who understands and follows my seven simple steps can unleash his or her inner math nerd, and Calculus will be a piece of cake.