If a function is analytic, this means:

1, the function can be differentiated infinite times, and its Taylor series at a point “c” in its domain converges to the function it self for some x in (c-R, c+R). In other words, a analytic function can be written as

,  for all x satisfying

2, the quality mentioned in (1) must be applied to all “c” in the domain of f(x)

Here are some examples:

1, all absolute value functions are not analytic. For instance, f(x)=|x| is not analytic since at  x=0, it is no differentiable, which does not satisfy quality (1)

2, Any function that has some point a where is always true is not analytic except f(x)=0. Since all of these functions’ Taylor series at x=a must be   thus the series fails to converge to the function when  x values are not zero

There are actually no technically difficult questions but ones which require carefulness and patient. What I mean are two specific cases:

The first case, in my opinion, may be the hardest one in the first question. since it has two different characteristics mixing which makes me confused of which technique I should choose. The first impression it left on me was that a normal substitution could not work since the numerator of the function is 1, so I try a trigonometric substitution and totally did twice substitution to finish the problem. However, it turns out that a normal substitution can work and even works more effectively, this surprises me when I redo this question.

The second case is actually very easy but tricky enough to let a student lose some marks since when solving the integration equation and get a function whose derivative is n times itself, most students only come up with a e^x model of f(t), which is generally correct except for the lack of a coefficient. Therefore quite a lot students missed a coefficient of f(t) in their answer.

Study tips:

1, Improve the skill of integrating. In fact, most integration problems don’t require a large amount of thought but sophisticated skill since most of them have evident patterns indicating which technique we should choose. However, such a skill might not come out immediately without sufficient practice.

2, Go over again the way we deal with volume and work problem and form a good habit. Volume problem can be solved in two different ways and both of them have their cons and pros, knowing how they actually work and practicing make us decide which to use quickly during an exam; Work problem can be very abstract so a good way to start is to figure out all the physical stuffs first.