Question 1 : Not much to say about this one, people forget to differentiate the x. A quite important remark though is that on this question as well as in others, students don't fully read the questions and do not answer fully (e.g. here didn't write the equation of the tangent line).


Also, people didn't know how to compute cos(pi/4) and answered with numerical values, I tried to emphasize that numerical approximations are not desired.

Question 2 : People were confused, lots used that x->x^3 is increasing which is quite equivalent to f continuous.

Question 3 : Too many "the function has to change direction twice so its derivative must cancel twice", i.e. vague sentences.

Question 4 : Not many rigorous answers. In particular, a LOT of people answered this question using arctan, but to define such a function we already need that the range is R and that is given by computing the limits.

Also people tried to wildly do lim(ab)=lim(a)lim(b) when b does not have a limit.

I think only 1 or 2 persons had full points on this question, it was a big mess altogether.

Question 5 : People did inductions without needing it, not even using the induction hypothesis.

Question 6 : Quite good, few people actually justified that the derivative didn't have a limit.

Question 7 : Good again but too many people have shown that the derivative had a limit at 0 without checking that this limit coincides with f'(0).

Question 8 : Very messy, lots did not finally prove that x^3 is increasing, some did it the other way around (used x^3 increasing to prove x^2+xy+y^2) but there were some clever solutions.

Examples : x^2+xy+y^2 = (x+y/2)^2+(3/4)*y^2, the sum of positive terms.            or x^2+xy+y^2 = (x+y)^2-xy > 0 if xy<0
              x^2+xy+y^2 = (x-y)^2+3xy > 0  if xy>0