To do these problems you should only need a basic understanding of single variable calculus; this entails a strong grasp of essential concepts like continuity, differentiation and integration, and a familiarity with some of the central theorems, like the Mean Value Theorem, L'Hopital's Rule, and the intermediate value theorem, most of which are common sense. In other words, the prerequisite knowledge is very low, and there won't be any really obscure "tricks" that are more transparent to people with an extensive background in mathematics. If you have taken AP Calculus you should be able to do these.
For fun I am thinking of writing a calculus competition for high school and first year university students, and we could do a trial run on the x86 forums! (e.g. a mini competition with similar problems to ones that I am thinking of using). I also want to compare the quality of education the people here have in single variable calculus; the first question here was actually given to me on a calculus exam a long time ago. I had 20 minutes to complete it, and the scaling was minimal. So if you attempt the question here and get say, 7/10, then that would be a rough estimate for your grade in this kind of course (70%).
Here are the questions! There exist complete and short solutions to each of them (less than half a page typed), so if you are finding that your solution is extremely long or complicated, there is likely a more efficient approach. Also it is important that you read the questions carefully. Finally, I think these questions are quite fun to do because all that is needed is some abstract thought, not obscure knowledge or tricks, and their solutions all say something quite meaningful/profound in analysis. I guarantee you will feel good if you solve one.
Let me know if you would be interested in setting up a small competition, or being used as test subjects for my test prototype!
(I will provide some hints throughout the questions)
Question 1
Prove that whenever
is continuous and
, then
(
Rule's hints: Note that this is actually a stronger statement than the Fundamental Theorem of Calculus -- the function f(x) need not be differentiable. To clarify notation, the function has any positive real number in its domain, and it maps this number to some other real number. The closed interval a to b is a subset of the open interval from 0 to 1. Basically a and b are numbers between 0 and 1, and b >= a. Please do not get sidetracked by the notation, as the essence of the proof should be in noting that f is continuous on a bounded interval. But there are many possible approaches so please keep an open mind and just look at the equation. Also it is not essential that you use the FTC to prove this question, and this result does not trivially follow from the FTC).
I will post a new question every week if there is interest. Also it is highly unlikely that you will be able to find a solution to these on the internet.
