Comments for 18:

30 for 13, 40 for 14, 30 for 15

average of 65.7, high of 95

3-13 was okay, although some didnt show why the RHS was equal to the desired
value and many forgot to explain why it is okay to show the proof only for x1
and x2 and not xi and xj

3-14 tripped a lot of people up. Lots of people used the hint but did not show
where the hint came from. They also reached the conclusion
suggested in the problem but did not explain how that proves the original
proposition. This was a tough problem.

3-15 was pretty good. Some students tried to overcomplicate it by using that D
function and that didn't really work because they didn't account for the case
where you differentiate multiple times with respect to a particular variable.
The easy way is to just show that you can switch any two consecutive dx's at
will, and hence you can rearrange them to make whatever you want. Some people
showed this but did not fully explain why it works, and others showed it for
R^3 and tried to pull a "WLOG", but I docked points for that because it really
isn't that hard to prove the result for the general case.