The first exam

36 students took this 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

The alert student will notice a certain resemblance between the exam given here and the first exam given in my section of Math 421 last spring.

The Oxford English Dictionary records the first printed appearance of the word lazy in 1549. The OED states that the primary meaning of lazy is "Averse to labour, indisposed to action or effort; idle; inactive, slothful."

Problem 1 (16 points)
a) (10 points) A routine computation using integration by parts, although errors are easy! It will help in part b) if you get the correct answer. Of course, a small part of the purpose of part b) is to check your answer. Students were specifically asked to use the definition to compute the Laplace transform in this problem. Other methods did not earn any points, but a correct answer was eligible for points in part b).
b) (6 points) Careful use of l'Hopital's rule is a major part of justifying some of the computations of the "operational calculus".
Comment Several students got incorrect answers to a) and then proceeded to get answers to b) which, through garbled computation, led to 1/2. These students have deficiencies in integrity or ability or both, perhaps.

Problem 2 (14 points)
This is a standard ODE problem which can be solved with various techniques (several from 244, for example). I think this problem is a bit easier than the corresponding one in the previous exam I gave.
a) (10 points) 4 points for taking the Laplace transform of the equation. 4 points for solving the partial fractions problem and setting up the equation for the inverse Laplace transform. 2 points for "reading off" the solution correctly from the tables.
b) (4 points) The points are for checking that the initial conditions are satisfied. Of course, this should help check your answer (as part b) was intended to help in the first question).
Comments Of course one basis of the solutions of the ODE y´´-y=0 is {e^t,e^-t}. Another basis is {cosh(t),sinh(t)} which some students used in this problem. Students should know the relationshops between these bases (in particular, what are cosh and sinh in terms of e^t and e^-t). Students who used the sinh/cosh basis should know how to differentiate these functions, and should also know their values at, say, t=0. This was covered on the first day of class. The answer in terms of sinh/cosh is 5cosh(t)+4sinh(t)-2. I penalized students who wrote solutions in terms of cos(i t) and sin(i t): they had better show me that they knew what they were discussing in terms of standard functions.
I note also that several students wrote formulas for y(t) which obviously did not satisfy the initial conditions, yet, in part b), these students just wrote that the initial conditions were indeed satisfied. Perhaps this is a joke, but to me again this may indicate a serious deficiency of either ability or integrity (or both!). Would you want to hire a person who did something like that, or work with such a person?

Problem 3 (12 points)
Straightforward application of one of the translation theorems and the table of formulas. Essentially I tried to grade using the following guidelines: correct appearance of e^-2s would get 3 points. Correct statement of an applicable translation theorem earns 3 more points. Use of the translation theorem is 3 points, and finally, writing the (correct!) inverse Laplace transform is worth the last 3 points.

Problem 4 (12 points)
This problem is a bit more difficult than the corresponding problem in my old exam. I increased the point value a bit. 3 points for knowing that the Laplace transform of the convolution is the product of the convolutions (this is on the formula sheet, so you need to have instantiated the result with the appropriate functions). The result is NOT the convolution, as a number of students seemed to think, but is the Laplace transform of the convolution. The inverse Laplace transform of this product must be computed. So 3 points for realizing the correct form of the partial fraction decomposition necessary, 3 more for finding the correct values of the constants, and a final 3 points for finding the correct inverse Laplace transform.

Problem 5 (20 points)
This problem is worth one-fifth of the exam. I don't think I could solve this ODE easily without the Laplace transform, or even understand a mathematical statement of the problem very well without using the Dirac and Heaviside functions!
a) (10 points) Here taking the Laplace transform earns 4 points, 4 points for "massaging" the Laplace transform (mostly another partial fractions exercise), and 2 points for writing the inverse transform. If a serious error is made in this part which "trivializes" (makes much easier!) some or all of the successive parts of the problem, then I will not give points for those parts.
b) (6 points: 2 points for each part) I am happy to accept an unsimplified formulas in various intervals although that will make c) and d) more difficult. Especiallly interesting to me are students who told me that y(t) was not 0 before Pi/2: this ideal spring starts in equilibirum (consider the initial conditions) and nothing happens to it until Pi/2. It should not move at least until Pi/2!
c) (2 points) This is actually a very simple graph. You can get enough information by evaluating at 0, Pi/2, Pi, and 3Pi/2. Even if you didn't simplify previously, you should be "suspicious" about the graph.
Comment The "geometry" of the axes supplied was supposed to be a hint, also. I will not consciously try to mislead students by supplying ludicrously inappropriate coordinate axes in such a problem.
d) (2 points) Maybe this is the theoretical part of the exam, although maybe engineers should be concerned about shocks. That y(t) is not differentiable at only one value of t is subtle to me. Certainly the solution is differentiable everywhere except Pi/2 and Pi.

Problem 6 (14 points)
I gave 5 points for writing the symbolic linear combination. Then correct use of a RREF or other method to get a solution earns the remainder of the problem's credit, and I reserved (at least) 2 points for a clear statement of the correct solution. I do not believe that the New Brunswick RREF can be used to solve this problem.
To get a significant number of points, I need to be satsified that the student knows what a linear combination is, in the context of this problem. The problem is very easy (two lines or three?) with the use of Piscataway.
Comment I believe that sometimes computation can be an effective teaching and learning device, I don't like to subject students to large amounts of pointless computation, especially on an exam. If you find yourself doing that on one of my exams, it may be time to think about your methods.

Problem 7 (12 points)
Students should know and "manifest" what is needed to verify linear independence. Then the linear system has to be set up and solved. 4 points for knowing about linear independence. 8 points for manipulating the system correctly and showing that the functions are linearly independent.
Engineers You can indeed survive an exam in an upper-level math course where one of the questions has the word "Prove" in its statement.

Extra credit (5 points)
I gave 5 points to students who were able to take a "random" (?) matrix produced by Maple and convert it to RREF. Ample opportunities for retakes were offered.
Comment Only 29 of 36 students got 5 points this way.

The second exam

34 students took this 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

     Please

1. Do my problems, not those you invent.
2. I cannot read your mind or imagine your answers: what you write is what I will read (and grade).
3. If you do extensive computation, you are probably not doing my problem (see 1) or you are doing my problem inefficiently.

Problem 1 (12 points)
6 points for each part. I will read what was written. I can not guess what you meant to write. I required that what was written be responsive to what was asked. In part a), an important "If ... then ..." needed to be inferred from what you wrote (for example, "If [a certain linear combination is 0] then [all the coefficients are 0]". Or "The only way a certain linear combination is 0 is if all the coefficients are 0". I also accepted statements about none of the vectors being linear combinations of the others. But I read what you wrote, and tried to read it carefully.
I took 2 points off in part b) if the definition used AX=X and did not specify that X must be non-zero. This specification is important. If a definition using the equation det(A-I_n)=0 was given, there was no such risk. Again, I read what students wrote, and did not guess at what they did not write.
Comment Since we spent a major portion of the course on linear algebra, and linear independence and eigenvalue are principal ideas in what we covered, students should be able to tell me what the phrases mean.

Problem 2 (22 points)
a) (2 points) Take the determinant of A-I. You may compute this determinant in any way you like.
b) (2 points) You may have already simplified the characteristic polynomial in part a), or you can do it here. The roots should be obvious.
c) (4 points) You need to solve three (3) homogeneous systems of linear equations. But they all are rather simple.
d) (2 points) You are merely asked to write P and D, which you certainly should be able to do after parts b) and c). Verification of your statements occurs in the next few parts of the problem. If incorrect results from b) and c) are used correctly here, I gave full credit.
e) (3 points) You may find P^-1 in any way you like. The answer is easy enough to check, so I gave 1 point out of 3 for an incorrect answer. The exam I handed out, by the way, had two parts labeled d. That is fixed in the posted version.
f) (2 points) Compute the product requested.
g) (2 points) Compute the product requested. If you do not get a correct diagonal matrix, I gave no points.
h) (5 points) 2 points for setting up the requested relationship A⁶=PD⁶P^-1. 1 point for computing D⁶ and 2 points for computing A⁶ (information was given allowing you to check your answer).

Problem 3 (16 points)
a) (8 points) A restatement of the definition in the language of the problem (that is, writing an arbitrary linear combination of the functions of the problem, setting this equal to 0, etc.) earns 2 points. The balance is earned when the answer contains verification that the coefficients of the linear combination must be 0.
b) (8 points) The correct answer (No) gets 2 points. The cover page states, "An answer alone may not receive full credit." Correct supporting evidence is needed for the other 6 points.
Comment Please see the lecture of October 5 for analysis of a similar example. There are also similar examples in some review problems.

Problem 4 (12 points)
In this problem, I expected students to evaluate the determinant (8 points). A few students tried other strategies (using RREF), and one or two of these students were successful. Students who "plugged in" values for a and b and c were simplifying the problem too much and can earn at most 3 of these 8 points.
I hoped the determinant evaluation would be combined with the knowledge that a matrix is singular (not invertible) exactly when the determinant is 0 (2 points). This then could be matched up with the perpendicularity condition to get the desired conclusion (2 points).

Problem 5 (18 points)
It is possible to make mistakes in a) and have serious effects on work in b) and c). If the result of a) were as complex as the correct answer, points were only taken off in a). A similar approach was followed for errors in b).
a) (5 points) One application of integration by parts. Keep track of the n's and the signs.
b) (4 points) Evaluate the antiderivative. Notice that sin(nPi) and sin(n 0) are 0 and that cos(nPi)=(-1)ⁿ and cos(n 0) is 1. I took off 2 points for errors which really fouled up the answers in c) (examples: omitting the sign "flip" or omitting the Pi).
c) (2 points) The points were earned if the result of b) was used correctly.
d) (7 points)
The left-hand graph (4 points)
In the graph of the partial sum, I looked for the following qualitative behavior:

1. Continuity on [0,Pi] with matching values at 0 and Pi
2. The value 0 at both 0 and Pi.
3. Closeness (with "wiggling") to the line segment inside the interval.
4. Gibbs phenomena (overshoot) at both ends

1. Identical to x+1 except at the ends (1 point)
2. 0 at both ends (1 point each)

Problem 6 (12 points)
With orthogonality, there is almost no computation in this problem (yes, other than small integers). I'll take off 1 point if the normalization constant is misquoted (I did this in my own solution of the problem, so I would have scored 99 at most). If the integrals are computed as the sum of squares of the coefficients with no constant (or, better, with the constant=1) I will take 2 points off.
Some students antidifferentiated instead of differentiating, and this will lose 2 points.
I took off 3 points for an error I didn't anticipate, an answer which essentially declares that the integral of (a negative number)² is negative. I can't read people's minds, and I don't know on what level this error was made: through fatigue and nervousness under exam conditions, or because of serious misunderstanding.

Problem 7 (8 points)
I am not satisfied with the statement of this problem. The statement might have made the problem more difficult for students. I wanted to ask: what is the polynomial formula when x<0 for the odd (respectively, even) extension of the polynomial x+x⁴? I think now I should just have asked exactly that. The statement of the question(s) seemed to invite misinterpretation.
4 points for each part, with 2 points for the specification of x<0, 1 point for x>0, and 1 point for identifying which Fourier coefficients must be 0.
I'll give 1 point on each part to people who correctly write the Fourier sine (respectively, cosine) series for F(x) (respectively, G(x)).
I'll put a "cleaner" version of the question in the posted exam, but I'll also show the wording I actually used, since I believe that weakened student efforts on this problem.

Extra credit (5 points)
I gave 5 points to students who presented answers to some questions about matrices in block form.
Comment Only 16 of 34 students got 5 points this way.

The final exam

31 students took this 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:

Problem 1 (10 points)
2 points for getting the e^-2s as a result of U(t-2). Then 2 points for rewriting the remainder in terms of t+2. 2 points for "expanding" (t+2)³ correctly, and 2 points for clarifying the exponential as e¹⁰e^5t. Finally, 2 points for writing out the complete answer.

Problem 2 (22 points)
a) (14 points) 4 points for taking the Laplace transform of the ODE. 6 points for changing it algebraically (including partial fractions) into a more manageable form. 4 more points for the inverse Laplace transform.
b) (2 points) 1 point each for correct formulas.
c) (4 points) 1 point for y(0), 2 points for the correct derivative, and 1 more point for y´(0).
d) (2 points) 1 point for the correct answer, and 1 point for some reasonable explanation.

Problem 3 (18 points)
a) (6 points) A correct definition of linearly independent should be given. In particular, I again looked for an appropriate "If ... then ..." statement (or one which could be recognized as such).
b) (12 points) I looked for a verification that the given functions were linearly independent. There are various ways to do this while considering a linear combination of the functions. For example, we could consider special values of x (such as x=1 and x=2). Or we could consider the coefficients of the quadraric polynimials and examine the coefficient matrix of the resulting homogeneous system.

Problem 4 (20 points)
Although the individual computations in this problem are easy, the chance of making at least one mistake while doing the problem seems large.
a) (2 points) Get the characteristic polynomial.
b) (3 points) Get the eigenvalues. It is easy to guess one root, and then "deflate" the results (lower the degree by one).
c) (4 points) Get eigenvectors for each eigenvalue.
d) (2 points) Write D and P.
e) (5 points) Find the inverse of the P declared in the previous section. The computation can be done either by row reduction or using the adjoint. If the suggested P and P^-1 are much simpler than the correct ones, only 3 of 5 points can be earned.
f) (2 points) Compute Z=AP. The points are earned by doing a correct matrix multiplication.
g) (2 points) Compute P^-1Z. No points are earned unless the result is D.

Problem 5 (20 points)
a) (8 points) 4 points for correctly using the supplied Maple formula, and incorporating the needed integral of 1. 1 point each for the first four terms of the Fourier sine series, simplified as requested.
b) (12 points) The left-hand graph (7 points)
In the graph of the partial sum, I looked for the following qualitative behavior:

1. Continuity on [0,Pi] (1 point).
2. The value 0 at both 0 and Pi (2 points).
3. Closeness (with "wiggling") to the line segment inside the interval (2 points).
4. Gibbs phenomena (overshoot) at both ends (2 points)

1. Identical to the function except at the ends (3 points)
2. 0 at both ends (1 point each)

Problem 6 (20 points)
a) (9 points) 2 points for separating correctly, with little or no other correct work. 4 points for writing correct X(x)'s and corresponding eigenvalues. 3 points for writing correct Y(y)'s. 2 more points for writing the product solution. If a product solution was not written, I also looked in part b) for a correct product.
b) (11 points) 2 points for writing a "formal" sum of the solutions gotten in a) with unknown coefficients. Then 3 points for setting t=0, and recognizing that the result is a Fourier sine series for the initial condition given, and using this information to compute the general form of the coefficient in the series. 2 points for using this coefficient in the series. 1 point each is earned for each of the first four terms.

Problem 7 (12 points)
The point (2,3) was written as (Pi/2,3) on the actual exam. Neither the students who took the exam nor the examiner noticed this! The picture, as was intended, supplied the initial data.
a) (6 points) 2 points for connecting the dots at the ends. The resulting graph should be a smooth function (2 points), below the initial conditions (2 points). 1 point taken off if the graph is above on one side.
b) (6 points) Again, 2 points for connecting the dots at the ends. 2 points for drawing an increasing function. 2 points for something that is straight or nearly straight.

Problem 8 (16 points)
a) (5 points) Write the D'Alembert solution for the PDE. No boundary conditions are given, so use of Fourier series techniques is not valid and earns 0 points. "Clarification" of f(x) by finding formulas for its pieces is not necessary.
b) (5 points) Sketch two pieces of the graph, at half the amplitude and the same width of the original shape, etc. If two separate bumps are given, 2 points are earned. If they are the same shape as the original profile, 1 more point is earned.
c) (4 points) The answer t=13.5 with some work shown earns the points. The answer 14 earns 3 points, and the answer 15 earns 2 points.
d) (2 points) 1 point for the correct answer, and 1 point for some reasonable explanation (it would be nice if the explanation included a word like "shock").

Problem 9 (18 points)
a) (4 points) Write the correct answer.
b) (2 points) The results are the same (1 point) because cosine is periodic (1 point).
c) (4 points) A useful and valid overestimate is desired. I looked for 15, with some explanation involving maximum values of sine and cosine.
d) (8 points) 6 points for some computation of the integral of (u(x,y,t))², somehow indicating how orthogonality is used. 3 points off if no evidence is offered. The max value earns a point, as does a value of t when this max value is attained.

Problem 10 (18 points)
2 points for recognizing that |x| is even so that the sine coefficients are all 0. 10 points for computing the cosine coefficients a_n for n>0, with all details. 2 points for computing a₀. 4 points for evaluating both sides of Parseval's formula as much as possible (this includes evaluating the integral).

Extra credit ( points)
23 students earned points by completing some part of the desired Maple graphs.

Course grades

Maintained by greenfie@math.rutgers.edu and last modified 12/23/2004.