The first exam

Numerical grades will be retained for use in computing the final letter grade in the course. Here are approximate letter grade assignments for this exam:

Discussion of the grading

Problem 1 (10 points)
6 points for describing the process of finding the 4^th roots. Some work should be shown because the cover sheet states, "An answer along may not receive full credit." If you showed some relevant algebra but no conclusive answer was displayed, I gave 3 points.
2 points for listing the roots in rectangular form, and 2 points for sketching these roots on the axes provided.

Problem 2 (14 points)
a) 2 points.
b) 4 points, generally divided 2 for the picture and 2 for the algebraic description.
c) 4 points, generally divided 2 for the picture and 2 for the algebraic description. Only 2 points were given if the second or fourth quadrants were indicated.
d) 4 points. Most people did refer to their answers to parts b) and c), although a few earned full credit by analyzing the problem algebraically. If the answer to part c) is incorrect, then the possibility of earning full credit here is certainly impaired. If the incorrect answer was the second quadrant, life was good. If it was the fourth quadrant, a good explanation of the assertion became more difficult.

Problem 3 (14 points)
I gave 2 points for correctly citing the Cauchy-Riemann equations, 2 points for differentiation the equation u=v² correctly with respect to x and y, and 2 points for combining the two sets of equations. I then gave 4 points for further work with the equations, yielding an equation with one partial derivative alone combined with u or v. Finally I gave another 2 points for correctly supporting the assertion that f(z) is constant. I did subtract 2 points if following an alternative correctly was not done. For example, if the equation u_x=-4v²u_x was obtained (as many people did) then either u_x is 0 or ...
I gave full credit to several alternative approaches which used valid mathematics.

Problem 4 (10 points)
Students who did not know the meaning of harmonic were severely hindered in this problem! I gave 5 points for finding the correct values of A This is where, as mentioned on the answer sheet, I made a mistake: A=0 is indeed an additional correct answer (I discovered only A=+/-3). I admit this, but didn't penalize those who, like me, discovered only A=+/-3. I also didn't reward those students who also found A=0. I am willing to discuss this decision with students. 5 points were earned for finding the needed harmonic conjugates. I required that some sort of verification or process be shown to support the claimed conjugates.
For both "chunks" of 5 points, I awarded 3 of the 5 if one alternative was correctly given.
It would have been better if the problem statement used the function e^Axcos(3y).

Problem 5 (12 points)
a) 4 points for log(-1) and 3 points for (-1)ⁱ. Of course the computations are connected. If there were not an infinite number of answers, only 4 points were available for this part of the problem.
b) 5 points. I gave 2 points for even mentioning the capital A variant of Arg. I also tried to be sensitive to students who might want to provide examples for a definition of Arg where the values are in [0,2Pi), not, as our textbook would have it, in (-Pi,Pi]. I did want "an explicit pair of complex numbers" so a discussion without giving a specific example did not receive full credit.

Problem 6 (10 points)
The simplest definitions for this problem (4 points) used the complex exponential function, and then the "proof" (6 points) is rather simple. I tried to give appropriate credit to students who used other definitions.
I penalized students 1 point for not identifying their definitions (I think this is a rather small penalty, since the definitions were specifically requested). I penalized students 2 points if exponential definitions were given without an i with the z. Then, of course, such definitions made verifying the stated identity much more difficult, or, in fact, impossible.

Problem 7 (12 points)
I gave 4 points for some parameterization evidence. I deducted 1 point for an incorrect final result, if everything else was more or less in good shape. I feel some sympathy for students who made small errors here since when I first did the problem in two ways (once by direct parameterization and once with Green's Theorem) my answers had different signs. I tried to grade so that small errors received small penalties.

Problem 8 (10 points)
1 point for citing the ML bound and 1 point for a correct value of the length.
4 points for treating the top of the fraction correctly. This means using the triangle inequality, and finding correct bounds for both the exponential function and the monomial.
4 points for treating the bottom of the fraction correctly. So here an estimate using the "reverse triangle inequality" needs to be given, with a correct selection of BIG and small when R is large.
1 point for putting together the estimates and 1 point for deducing the required result (with a limit).
I deducted 2 points for asserting that complex numbers can directly participate in inequalities. C is not an ordered field.

Problem 8 (10 points)
1 point was reserved for the final result, which could have been obtained in various ways. I looked for supporting evidence using, for example, the Ratio Test for the other 7 points. I also looked for the word/symbol, "limit/lim" appropriately.

The second exam

Numerical grades will be retained for use in computing the final letter grade in the course. Here are approximate letter grade assignments for this exam:

Discussion of the grading

Problem 1 (15 points)
2 points for a sketch of the contour.
6 points for the residue computation.
6 points for showing that the ML estimate makes the integral over the "other" part of the contour go to 0.
1 point for getting the answer.
3 points for the correct extra credit answer.

Problem 2 (14 points)
a) 7 points. simply and simple each get 1 point, properly used. The equation itself gets 3 points. The other "stuff" gets 2 points.
b) 7 points. The answer, and some method for getting it.

Problem 3 (15 points)
The singularity at 0 can earn 7 points. The other singularities can earn 8 points.
At 0, 1 point for the statement that the singularity is removable and 1 point for the value of the residue. The other 5 points are for some supporting evidence.
Similar scoring will be used for the non-zero singularities: 1 point for the residue and 1 point for identifying the singularity as a pole and 1 point for giving the order of the pole. Again, supporting evidence can earn 5 points.

Problem 4 (12 points)
a) 8 points. Apply Liouville's Theorem (2 points) to a "correct" function (4 points) and get the conclusion (2 points).
b) 4 points. 2 for the assertion that the exponential function does not have modulus bounded away from 0. 2 more points for explaining why.

Problem 5 (14 points)
2 points for information about a power series for sin(z).
3 points for information about a power series for 1/(z-1)².
5 points for combining them usefully (multiplication, division, etc.)
Then 4 points for each of the correct terms in the answer.
An alternative unrecommended direct approach is possible. So computing f^(k)(z) correctly will earn k points, where k is an integer running from 1 to 4. Thus computing f(0) itself earns nothing (sigh). Assembling the terms in the Taylor series (with the factorials) earns, as above, 4 points for each of the correct terms in the answer.

Problem 6 (14 points)
a) 4 points. Many correct answers are possible, and if another answer (inferentially) needs to be chosen to answer d), that's o.k. 2 points for excluding 1, and 2 points for making a correct "cut" in C. b) 2 points for the correct answer.
c) 2 points for the correct answer.
d) 7 points. Some derivatives of sqrt(z) will earn 2 points. This should be a series in integer powers of z-1. Each correct term will earn 1 point.

Problem 7 (15 points)
5 points for instantiating the conventional "dictionary" changing this to a complex line integral. Finding the singularities of the integrand and manipulating it algebraically correctly earns another 5 points. Applying the Residue Theorem, and computing the correct residue and getting the correct answer earns the final 5 points. The not-uncommon error of failing to compensate for the fact that the dictionary response does not get a monic polynomial will lose 2 points. This error usually yields an extra b in the answer.

The final exam

Here are approximate letter grade equivalents. I think the final exam was not easy.

Discussion of the grading

I did not prepare an answer key for the exam (one of the few pleasures of giving a final exam!). I did do the problems, though, and tried to grade carefully. The rules of the university state that I should retain final exams, but I would certainly be happy to look at your work with you or even provide a copy of your work to you. I have some comments about the problems and about student performance.

Problem 1 (20 points)
One of the more straightforward problems.

Problem 2 (20 points)
The mean and median grades of this problem are a disappointment and somewhat of a surprise, especially contrasted with, say, problem 7. I spent 1.5 periods directly on this material. Oh well, this was the end of the semester. Please see the lecture notes from the last lecture for a solution.

Problem 3 (20 points)
Routine. I should mention that as stated in its source, the problem asks for fourth power of x²+2x+2. I decided the square would be sufficient!

Problem 4 (20 points)
This problem looks weird and difficult, which is why I provided what I hoped would be a useful hint. The problem is nearly straightforward, once the shock of the statement is gone.

Problem 5 (20 points)
This is an imaginative problem. I wanted people to explain why there would be only one reasonable analytic continuation of x^x, and that the formula would involve Log and Arg.

Problem 6 (20 points)
This to me is quite straightforward, once one gets over the shock of the statement of the result. g(z) is just f(z) minus the beginning of f's Taylor series, divided by the appropriate power of z-a.

Problem 7 (20 points)
I spent less than half a period on Rouché's Theorem, yet people did well on this problem.

Problem 8 (20 points)
This problem shows that the max of the function controls the max of the derivative in a very direct way. This relationship is one of the true weird results of complex analysis with no analog in calculus.

Problem 9 (20 points)
A routine problem. The words Liouville and exp would get almost full credit!

Problem 10 (20 points)
One of the standard ways to define Bernoulli polynomials (and others!).

Course grades

Maintained by greenfie@math.rutgers.edu and last modified 5/9/2005.