### The final exam

Problem #1 Problem #2 Problem #3 Problem #4 Problem #5 Problem #6 Problem #7 Problem #8 Problem#9 Problem#10 Problem#11 Total 18 18 12 18 12 16 16 14 19 18 12 171 5 4 0 4 2 1 3 0 1 0 0 53 12.42 14 7 13.46 9.69 9.62 10.92 7.73 8.27 9.92 8.73 111.77 13 15.5 8.5 14 12 9 12.5 8.5 7.5 11 10 107

26 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:

 Letterequivalent Range A B+ B C+ C D F [147.9,174] [139.2,147.9) [121.8,139.2) [113.1,121.8) [104.4,113.1) [95.7,104.4) [0,95.7)

Generally arithmetic errors will be penalized only minimally. If, however, your error makes the problem much simpler, more credit will be deducted and your answer may not be eligible for all of the credit of the problem.

Problem 1 (18 points)
a) (13 points) 3 points for the setup, which includes the correct description of f(t) and the correct domain of integration. 4 points for the integration by parts. 3 points for evaluation of one piece of the integral with 1/s and 3 points for the evaluation with 1/s2. If I could not easily understand the computations, I gave up. I think part of this problem is communication of the limits of integration, the methods, etc. If the setup was f(t)=2t for t<1 and 0 otherwise, a maximum of 10 out of 13 points could be earned.
b) (5 points) 1 point for demonstrating the need for L'H once, and 1 point for correct top and bottom differentiation. Another 1+1 was earned for the second L'H. The final point was earned for the correct answer. I was rather strict here. Students had to show why L'H was needed, and had to compute correctly. One consequence is that answers from a) which were far from true led to ineligibility for the points in this part. That should have been a signal to students that their process and/or answer in a) was incorrect.

Problem 2 (18 points)
a) (12 points) 3 points for taking the correct Laplace transform of the equation and solving for Y(s). 3 points for the partial fraction decomposition. 3 points for the inverse Laplace transform of the rational function. 3 points for the inverse Laplace transform of that part involving the exponential function (one of the Shifting Theorems is needed). Students who recognized the Laplace transforms of sinh and cosh and used them correctly got full credit. I would have put these on the table if anyone had asked (but then I would not have given this precise problem!). Students who made errors which trivialized the problem were penalized here, and might not have been eligible for full credit in the parts below.
b) (3 points) 1 point for the first expression, and 2 points for the second.
c) (3 points) 1 point for the first answer, 1 point for the second, and 1 point for the third. Students needed to use their answer, and compute correctly.

Problem 3 (12 points)
a) (6 points) The most common error was omitting the condition that v not be 0. I gave credit only when I could understand the statement and it was correct.
b) (6 points) An answer alone does not get full credit. A maximum of 2 points would be given to an answer supported by very little. I looked for assurance that the student had checked that every coordinate of the vector gave the same eigenvalue.

Problem 4 (12 points)
a) (10 points) The resulting polynomial should have degree 4, or the student loses 2 points. Most students "expanded" along a row or column, and I subtracted 2points if one of the resulting 3-by-3 determinants was incorrect. Other minor errors were handled appropriately.
b) (2 points) For displaying the correct roots of the student's characteristic polynomial. Almost all students got polynomials whose roots could easily be found by hand (thank goodness!).
c) (4 points) To score points in this part of the problem, statements needed to be supported by previous student work, and by valid reasons for the conclusion. Therefore I did not give points to someone who might have guessed the correct conclusion but was unable to give some valid supporting evidence.
d) (2 points) The correct answer (1 point) and some correct reason (1 point).

Problem 5 (12 points)
a) (6 points) I gave 2 points for considering a linear combination with "arbitrary" coefficients and then setting it equal to 0. The resulting homogeneous system was easy to analyze and show that it had only the trivial solution. I did not give credit if I could not understand what students wrote.
b) (6 points) The part of this question certainly is the worst constructed from the point of view of checking if a student knows the subject matter of 421. That is because it is clear (at least, I think, to me!) that a sum of polynomials whose degree is at most 2 cannot be x3. (It is clear, however, because all of us have done so much computation in the "native" basis of the polynomials, that is, the usual monomials.) Therefore any student who stated such a belief in a way I could understand got full credit. Thus those students did not have their 421 linear algebra knowledge tested. Otherwise, I deduced points if students didn't describe the problem clearly. In particular, I gave only 4 points if students asserted that x3 was not a linear combination of two of the three given functions, or if they asserted that x3 and two of the given functions were linearly independent. Either of those is not enough to conclude the desired result.

Problem 6 (16 points)
a) (8 points) I gave this problem so that people could show me they knew what Fourier coefficients and Fourier terms were. There were rather low scores on a similar problem on the second exam. Here I wanted the terms, and not just the coefficients, as I indicated. Any student who gave a linear combination of sin(x), sin(2x), and sin(3x) with other than constant coefficients got 0 for this part. By now, students should know what Fourier series look like. The 2/Pi normalization got 1 point. Each Fourier coefficient got 1 point (3 points total). Each appropriate sine function got 1 point (3 points total). 1 more point was given for assembling the information and presenting it correctly.
b) (8 points) 4 points for the graph of g(x): 1 point each for sticking to the continuous pieces "inside" the intervals of continuity; 1 point for being continuous in the whole interval and "leaping down" from the top to the bottom; 1 point for the Gibbs wiggling in both places. 4 points for the graph of h(x): again, 1 point each for being identical to f(x) on the two intervals of continuity; 1 point for having a value in the middle of the jump, and 1 point for noting in some way that there are holes on the bottom and the top of the jump. I deduced points for extra information which was contrary to the correct pictures.

Problem 7 (16 points)
a) (3 points) 1 point for the graph=1 away from the interval from 0 to 2, and 2 points for the parabolic bump up inside the interval.
b) (3 points) Just quoting the D'Alembert solution (with no g term!) earns the points, with the implication being that the function f is the one defined in the problem. Students who went on to state an incorrect formula (that is, substituting the quadratic formula with no indication of the valdity of the formula in terms of the values of the variables) lost 1 point.
c) (4 points) 1 point each for the traveling wave to the left and the right; 1 point for the correct height and location; 1 point for the wave being 0 outside of intervals of the traveling waves.
d) (1 point) For identifying the correct time.
e) (4 points) 2 points for indicating that there are 2 maximums. 1 point each for the correct location of the maximums.
f) (1 point) For identifying the correct velocity.

Problem 8 (14 points)
a) (4 points) I looked for a solution which satisfied the initial conditions (2 points) and which had the desired exponentials (2 points). The solution could just be written and not justified.
b) (8 points) I gave 2 points for some evidence of orthogonality displayed. Otherwise, I looked for the double integral, I looked for orthogonality, I looked for the appearance of the normalization constants, and I looked for the t functions not being integrated. Points were taken off appropriately for algebraic errors. etc. In this part, an answer alone got little credit: I looked for some supporting evidence.
c) (2 points) The answer, supported by some evidence derived from the answer to b). I gave only 1 point to thos students who insisted that the temperature oscillated, even if supported by b)'s answer. That's just too unphysical.

Problem 9 (20 points)
a) (9 points) 2 points for separating variables correctly. 2 points for handling the x function and coming up with sin(nx), where n is a positive integer. 2 points for getting the differential equation for Y(y) correct, and realizing that solutions are exponentials or hyperbolic functions. 3 points for explicitly listing the solutions (the case n=1 requires separate comment).
b) (11 points) 2 points for excluding the sinh term. 3 points for writing the correct series. 2 points for explicitly writing how u(x,0) is related to the initial condition. 2 points for writing u(x,0) as a sum of a Fourier sine seres. 2 points for writing the formula for the coefficients and 2 more points for writing the three initial terms.

Problem 10 (18 points)
I gave a total of 10 points for the separation of variables portion of this problem. The t and x portions were graded in the same way: 2 points for getting the correct ordinary differential equation, 2 points for getting cos(n{x|t}) (not a general ) and 1 point for dropping sin(n{x|t}). 8 points for assemblling the general answer. Of this, 4 points for a sum of the solutions with coefficients (ancos(nx)cos(nt), summed from n=0 to infinity) and then 4 more points for connecting with the initial conditions: first, for using u(x,0) in connection with f(x), and then writing the coefficients an in terms of cosine Fourier coefficients of f(x).
Comment This is about what's called the free boundary condition for the wave equation. Some further information is here.

Problem 11 (12 points)
a) (6 points) The curve should be smooth and higher than the global min, and lower than 3. It should have value 2 at x=0 and should have a horizontal tangent at x=Pi. I gave 2 points for the smoothness, 2 points for the horizontal tangent, and 2 points for going through (0,2). Sometimes the horizontal tangent at x=Pi was difficult to score.
b) (6 points) A horizontal line earned at least 3 points. 2 points were earned for a graph going through (0,2). The solution should be a straight line (the only steady-state one dimensional heat equation solutions), which accounted for the other points.
Here are some pictures of an approximate solution (with 100 Fourier terms). The correct eigen{function|value}s are given as the third example here. I used them to create these graphs, after first subtracting the appropriate steady-state solution.