Remarks :
Exercise 1 : The only notable mistake is when using MVT, some people take the midpoint where they want, when it's the opposive, there is one middle point on which we have no control which satisfies the mean value. And MVT works this way, not the other, you shouldn't start from one point, and find 2 others such that the first point satisfies the mean value, that is not true in general.


Exercise 2 : Three main mistakes here. Some inductions are still sloppy, you need to prove the initial step, and then suppose you proved it for "some" k, and prove it for k+1. Forgetting the initial step, or taking a particular k are both wrong.

Also when you have to find polynomials, it is better to say what those polynomials are explicitely, it sometimes feel like the exercise was stopped in the middle of a calculation.


When you had to compute the derivative at 0, you had to prove that it exists, that is to say we are a-priori not sure that it does, so you cannot just show that the limit of f'(x) when x->0 is 0, we do not know whether the derivative exist therefore we know even less if it is continuous at 0. In those case you have to use the definition of the derivative, with the limit of the quotient.


(Also sometimes the (e) was not attempted, it was the only subquestion with no hint, but it was still assigned).


Exercise 4-5-6 : It was good in general, sometimes informations were missing, make sure you gave everything that was asked. Also one should avoid writing arccos(-1) or arcsin(0), because for one, we know what those are, and it only give values in [0,pi) or (-pi/2,-pi/2], not all the possible are obtained.


Exercise 7-8-9 : When you have to approximate, try to do it only once, at the very end. Until the end, keep everything exact. This caused in a few occasions that in a 3-step calculus, the result was rounded at each step and the final result was significantly different from the actual one. (I didn't take points away but it is something to keep in mind for a possible scientific career).