I was asked to do more deeper question related to question 2 in the assignment.

For instance f(x)=sin(x) and g(x)=e^x, f(x) is analytic at c, and g(x) is not.

A function analytic at c means this function convergent power series for x=c. And only if Taylor series for particular function converge to this function for every single c at x=c.

Analytical functions are supposed to be infinite differentiable. It does not mean all the infinitely function is analytic, such as y=e^(-c)cos(cx).

The reason for its usefulness is that if we know that a function is analytic when x=c, we can get power series at x=c in domain.

I did really bad in this midterm. I was not careful about the webwork question and in classes I seldom took notes in this first half of term. My tips is obvious that we are supposed to do all the webwork problems carefully and listen to big and lecture carefully. Be sure to take notes in class and review them later.
The last question of first problem is the most difficult question of the whole exam. The question is evaluate integration 1/t^2sqrt(t^2+1) from 1 to sqrt(3). The reason is that two substitution are needed for the question. And we have to be familiar with sec and tan which we did not always use it in high school.
In the last half of the term, I will try hard and get a good marks in the final.