### The first exam

Problem #1 Problem #2 Problem #3 Problem #4 Problem #5 Problem #6 Problem #7 Problem #8 Problem #9 Total 12 10 10 12 12 12 12 12 8 99 2 0 0 0 6 0 2 4 0 31 9.75 5.94 5.81 7.88 10.63 7.56 10.31 10.13 5.88 73.88 11 7 8 9.5 12 6.5 12 11 8 74

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 [85,100] [80,84] [70,79] [65,69] [60,64] [55,59] [0,54]

### The second exam

Problem #1 Problem #2 Problem #3 Problem #4 Problem #5 Problem #6 Problem #7 Total 15 14 12 16 12 15 16 98 8 10 4 5 4 3 4 51 13.13 13.33 11.33 10.07 10 10.53 14.87 83.34 14 14 12 8 11 12 16 85

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 [85,100] [80,84] [70,79] [65,69] [60,64] [55,59] [0,54]

I misstated the point values of problems 1 and 2. I wanted #1 to be worth 15 points and #2 to be worth 14 points. That's a 1 point shift, and that's how I graded the exam.

The Lagrange multiplier problem was difficult to grade. Credit is difficult to give when logic is not clearly shown. I gave 4 points for the setup (the multiplier equations and the constraint equation). This problem has many "candidates" for where extreme values can be found. One collection (the points (+/-1,0,0)) come from setting lambda=0. This was worth 1 point. The collection (0,0,+/-1) was worth 2 points. The four candidates with z=0 (where the minimum occurs) was worth 3 points, and the candidates gotten by solving a quadratic (in either x or the multiplier -- I saw both) were worth another 3 points. I then gave 2 points for the correct answers (which had to be the values of the function!). Finally, I gave 1 point to the perhaps subjective evaluation of the logical clarity of the presentation. Perhaps I should have weighted that more heavily, but then I'd have to worry about partial credit. A few students did the problem in a non-routine fashion, and I tried to score their work consistently with what I described above.

The other non-routine problem was #6, involving Green's Theorem. Here I also found difficulty in understanding some solutions. There was sufficient time to explain what was written, I believe. Some students might have lost points because I could not understand their assertions. I wanted to see explicit mention of how Green's Theorem was used, including description and evaluation of the "bottom" integral. I wanted to read some reason that the integrand in the double integral simplifies greatly.

### 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 20 18 18 20 20 16 20 16 16 16 20 198 11 9 0 5 4 7 14 2 0 0 14 100 18.33 15.27 15.6 11.9 16.13 12.33 17.67 15 7.2 10.87 19.13 159.3 20 17 18 10 19 12 20 16 8 12 20 163

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 [170,200] [160,169] [140,159] [130,139] [120,129] [110,119] [0,109]

The problem which was most difficult to grade was #4. I expected this, because the question had no hints, and, since it used no "calculus" yet also asked for a creative solution, it generated a very wide variety of responses. The cosine called for, by the way, is indeed 1/3. I don't know the angle in terms of "standard" angles. I've been told that methane's molecule (CH4) is a regular tetrahedron. You may know that there are exactly five regular polyhedra with all faces congruent: these are called the Platonic solids.

The problem whose results surprised me the most was #9, about the chain rule. The chain rule is an important and difficult result in in several variable calculus, and a question about it probably should have been expected on the final. I thought the question I asked was not as difficult as others on the same topic which students in this class were already asked. I found the poor scores on this problem disappointing.

Of course the curvature problem was also non-routine, but most students did o.k. on it. I tried to apply an unbiased grading scheme, which is somewhat difficult in such qualitative problems. In #12, the weird double integral, I wanted some reason that the sin(x3) term would disapper, and I penalized people a tiny bit who didn't give a reason or gave a wrong one.