A function being analytic at a means the Taylor Series for the function at a converges to the function itself in an open interval centered at a. Taylor Series fails to describe what happen at the point a and the inside of the function. In question 2, the function is not analytic at a=0 because the Taylor Series at a=0 (the Maclaurin Series for it) is not convergent to it.

For example, function y=x^2 is analytic at any points on its domain. Its infinitely differentiable and its Taylor Series at every point is convergent to x^2.

Another example for non-analytic function is y=|x|. It is not analytic at x=0. Because its not eifferentiable at that point , we cannot even write a taylor series for it.

This concept is important since calculus aims to describe the inside of a function and it is both necessary and interesting to know when it does not work.

All the analytic functions must be infinitely differentiable but not all the infinitely differentiable functions are analytic.

In my opinion, the most difficult question on the midterm is Question 1. Although Question 1 is considered as the most basic fundamental computational skills that we need to master before talking about other more advanced calculus problem that demand more thinking or different kinds of thinking, the question took me 60 minutes in a 90 minutes’ exam. I thought that as long as I “knew how to do” the computation, I would be able to do them quickly in an exam but it turned out that i was wrong. My computational speed did not fit the requirement. For other questions, the ways of thinking them were already covered in assignments or other materials, which made the thinking itself not hard under an exam condition, and underneath all that thinking , is still, computational skills, that actually calculate the answers!

Tips:

1. Mathematics learning does need practice! Do not think it is sufficient to just “know how to do it”. The fact that the level of education changes that I am not in high school anymore and I am now doing things that are cooler than just computation does not change a core factor of mathematics–computation.
2. I should have managed my time more reasonably in the midterm and spent more time for the other questions.