### Discussion of the final, spring 2002

Thirteen students took the final. The mean grade was 154.07 and the median was 149. The high grade was 182 and the low grade was 95. Thus you can see that statistically this exam's results were qualitatively different from the other exams, where the median was higher than the mean. The exam was somewhat more difficult than anticipated. The instructor "took" the exam twice. His first attempt was done a week after writing the final, and his second attempt another week later. Both times he got some small detail wrong. His grade would have been 198 out of 200 (ignoring the possibility of the extra credit points).

Problem
number
Mean
score
Median
score
Minimum
score
achieved
Maximum
score
achieved
Discussion
120.0020 2020 Students used the Cauchy Integral Formula for the second derivative or the Residue Theorem.
215.0816 820 The problem, although routine, had the square root. Some people experienced difficulty. I looked for estimates and graded straightforwardly.
315.5417 420 A few student responses were far from correct. I gave much "partial credit" to students who made an arithmetic error -- the usual errors yielded a wrong coefficient on the second Laurent series term which was slightly irritating to compute correctly.
417.3118 620 I thought this would be a straightforward problem. In retrospect, I would have changed the second sentence of the problem to be more precise: "Write the resulting complex analytic function u(x,y)+iv(x,y) as an analytic function of the complex variable z=x+iy." I did give credit to students whose answers contained (in my opinion) unnatural (!) formulas involving z. (You should have read my mind in this question!!).
516.6919 520 Of course, this problem can best be done with some version of Rouché's Theorem. I think the problem must be done in two stages, first finding the number of roots inside the unit circle and then finding the number of roots inside the circle |z|=2. Some students persisted in writing inequalities with non-real complex numbers.
616.8518 820 This problem was straightforward for those people who didn't get too anxious. All the details fall into place. Some students who made a lucky mistake in part a) were actually able to get a correct answer in part b) because the imaginary parts of exp(ix)/(2-exp(ix)) and 2/(2-exp(ix)) are the same!
710.9210 518 This was one of the hardest problems on the exam and its median grade was the lowest by a large margin. One reason might be that conformal mapping was the last major topic covered in the course. We didn't have much graded homework on it. The method used for one part of the problem was discussed in class and in the textbook, but I don't think any graded homework was given about it. Here I am referring to conformal mappings that change angles on the boundary of a domain. So an angle changing map (square root in this problem) is needed. Perhaps I should have given a better hint. I did not give more than about half credit in the problem to students who essentially asserted that a correct solution could be obtained with just a linear fractional transformation.
816.1518 720 In part a) I wanted a specific citation of a result. For example, the citation I expected was Cauchy's Theorem, although several other results were correctly cited and applied by students. I also wanted to know where Log z was analytic in order to apply the result.
Almost everyone got part b) right, but I did want answers "in rectangular form".
916.2318 1020 I looked for discussion of the singularities of W(z), for evidence that the singularities were of the type described, and, finally, for some statement or reasoning connecting the type of the singularities with the radius of convergence given as the student's answer. A few students seemed to pay no attention to the Comment which was meant as a hint.
108.6913 020 This was, measured by its mean grade, the most difficult problem. I was reminded of discussions with people who wrote the AP Calculus exam. They said students for years had assumed the last free response question was always too hard, and that there was great confusion in the community when, a few years ago, the committee in charge of the exam essentially announced that an effort would be made to have the last problem not be the most difficult.) I believe students answering this exam were tired and perhaps additionally might have a similar fear of the last problem. I didn't want the problem to be very difficult.
I tried to identify the following features in student answers:
 i) An analytic function defined in a disc is equal to the sum of its Taylor series centered at the center of the disc (here the center is 0). ii) The sum of a power series is analytic inside its radius of convergence. iii) The Taylor series of F(z) converges absolutely for all z, and therefore converges for all z: a complete and correct discussion of this fact likely would refer to "Absolute convergence implies convergence" and the fact that the hypotheses on the derivatives of F(z) allow the conclusion that the sum of the series is overestimated by a series with positive terms which converges for all z.
I wanted my grading to be consistent and to reward correct effort. I looked closely for some statements of all of these ideas (or at least words I could interpret as such statements!).
In looking at the extra credit efforts, I wanted to identify exactly where "connected" was used. I did not award any extra credit points.

There were relatively few lost "it" referents in the answers to the final exam and I thank students for this. I do not thank students for sentence fragments whose meanings I had to guess.