How to approach odd, intimidating, and grotesque calculus problems

Adapted from G. Polya’s classic “How to Solve It” (1945)

1. Understand the problem in your bones (feels like lab work)

a. Read and re-read the problem.

b. Can you draw a picture or make a rough sketch?

c. Can you write out the first few terms of the series?

d. What exactly is the question asking? Do you believe it?

e. Can you break a complicated problem down into smaller, easier problems?

f. For abstract problems, can you think of a simple example to gain intuition?

2. Devise a plan – find a path to the solution (feels like brainstorming)

a. What is the connection between the data given and the unknown?

b. Have you seen the problem before?

c. Have you worked a similar problem?

d. Is there a useful theorem?

e. What are other problems with the same unknown?

f. Suppose you think of a related problem that you have solved before…

i. Can you use the result (like the convergence of a known series)?

ii. Can you use the same method (like a method of integration)?

g. Can you restate the problem? How many ways can you restate it?

h. Is there a slightly simpler problem that you can solve? Do it!

i. Is there a more general problem that you can solve? Do it!

j. Can you drop part of the conditions and solve the easier problem? Does this help?

k. Can you add more data to the problem and then solve it? Does this help?

l. Have you used all the information given?

m. Once you “see” the solution, find a logical and coherent plan to explain your reasoning.

3. Carry out the plan (feels like accounting)

a. Write out each step carefully.

b. Check your work as you go, so mistakes don’t propagate.

4. Reflect on your solution (feels like philosophy)

a. Does your answer make sense?

b. Is there a fast way (like taking a derivative) to check your answer?

c. Is your answer what you expected from step 1?

d. Is there something deep or surprising about your solution?

e. Bask in the glory.