These are three “Coffin” Problems posed by Nakul Dawra on his Youtube site GoldPlatedGoof. (Nakul is extraordinarily entertaining and mesmerizing.) The origin of the name is explained, but basically they are problems that have easy or even trivial solutions—once you see the solution. But just contemplating the problem, they seem impossible. The idea was to kill the chances of the pupil taking an (oral) exam with these problems. I was able to solve the first two problems (after a while), but I could not figure out the third. See the Three Coffin Problems.
For problem 1 Jim says: “I kept slipping into assuming what I wanted to prove”.
I think that has also happened on problem 2 (page 5 of the linked pdf), the Calculus solution.
He writes down the integral form, swaps the limits to get the required reversal of sign, then discards the limits on the differentiation step. Effectively a minus sign has been (happily) created.
If he differentiated, without swapping the limits first, the answer would be quite different. Looks dodgy to me.
Thank you so much for the comment (one of 14 out of approximately 10,000 spam comments I have gotten over the 3 ½ years for the blog).
First, the swapping of limits on the definite integral with the attendant switch in signs is legal. I did it to put the integral in the standard form for differentiation with the variable as the upper limit and the constant as the lower limit. Then the derivative of the integral is just the integrand evaluated at the upper limit (one form of the Fundamental Theorem of Integral Calculus). Things often look opaque when presented in an unfamiliar setting.
Thanks again for reading the blog and commenting.
Incidentally, I “nicked” part of your solution to Problem 1 because it was so elegant.
I have written up my misunderstanding of the (valid) limit swapping as a new problem as it could be instructive to others.