As we are talking about functions, we know that there are functions that are infinitely differentiable. An analytic function is a function that its Taylor series at any point x_0 in its domain converges to the function itself for x of x_0. In other words, if a function is analytic at c, then the function can be expanded to a  power series around c  has a positive radius of convergence. For example, if f(x) = e^x is analytic at x=0, then we are able to write out the power series representation ∑((x^n)/n!) as n≥0 near 0. So, the major difference between analytic functions and infinitely differentiable functions is that we can always write out a power series representation for analytic functions but not every infinitely differentiable function has a power series representation on every x in its domain.

The most difficult question on the Midterm to me is part (b) of Question 4, which is to prove the integral of f”(t)cos(t) dt ≤ f'(2π) – f'(0) from 0 to 2π. When I was doing the question, I was able to prove part (a), so I assumed that there may be some connection between the two parts. Eventually, I just could not find any tips of proving it. However, after redoing it and finding help from others, I noticed that I should use integration by parts twice in order to prove the problem. The way of solving part b is completely different from the way I solve part (a). So this part of the question is tricky and difficult to me, it teaches me not to try to solve a question in only one way.

Finally, I would like to present two study tips I recommend:

1.  First of all is to get used to all the techniques for solving problems that we learned in class because some questions on the exam may have only one way to solve. As we are familiar to all of them, there will be no worries when facing those questions.
2. Secondly, when we are studying for the exams, we should try to do the problems that we did not know how to do previously. Try to find help from others and redo those questions again and again until we get used to them. With that, the exams will not be nightmares to us.