The results from those of you who did the sample exam (and the fact that relatively few of you handed it in) indicate that there is considerable work yet to do. The following are some general comments. The solutions follow:

• Problems 1,2 are worth 5 points each. The others are worth 10 points.
• Problems 1 and 2 should be trivial. "Memorize" means memorize--really. (Then think about what the definitions mean.)
• Be careful what you say. I will grade the exam the same way you are judged by those listening to you in other contexts. If you say incorrect or incomprehensible things people will treat you accordingly. Employers and engineering professors won't dig through incorrect or silly statements to find something that is correct.
• Particularly offensive is when you contradict yourself. For example, if in one part of a problem you "show" an series converges, then in another "show" it diverges, you have said nothing. In fact, not realizing that you have contradicted yourself is worse than saying nothing.
• Make sure you write exactly what you mean. For example, the saying "limit (n to infinity) (2n+1)/(3n) = (2+(1/n))/3=2/3" is just wrong. The middle fraction is still a limit, so you must say "limit". You won't be sitting next to me when I grade the exams, so you won't be able to say "of course, I meant limit".
• It is OK to state a fact, even if you are not sure it is relevant--if it isn't relevant, you won't get any credit.

Some tips on studying:

• Memorize means memorize.
• Doing problems, first easy problems, then hard problems is the only way to get ready to do hard problems. (British students don't say they are "taking math", they say they are "doing maths".)
• Re-doing problems you have already done can be a useful way to practice writing-up skills. You already have some idea (from memory) how the solution will go, so focus on how to write the solution completely and succinctly
• Don't just re-read notes. My son's 4th grade teacher reacently told us about some research on how much people remember from different activities. A reader only remembers about 10 percent of what they read. Watching a performance (i.e., lecture) leads to 40-60 percent memory. However, if you "act something out" you remember 70-80 percent. So the most effective way to study is to first work on the problems yourself, then find someone who hasn't done the problems and help them.