### The first exam

Problem #1 Problem #2 Problem #3 Problem #4 Problem #5 Problem #6 Problem #7 Problem #8 Problem #9 Total 18 12 10 12 12 12 10 8 6 100 9 5 0 4 7 6 2 7 3 63 16.15 10.55 7.2 11.05 11.25 11.05 6.15 7.95 5.7 87.05 17 11 8 12 12 12 6 8 6 89

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 Green's Theoremfor a triangle Total 14 12 10 16 16 12 20 5 104 10 0 5 8 0 4 12 2 69 12.9 10.3 9.35 12.65 13.3 11.15 18.8 4.067 91.6 13.5 12 10 13.5 16 12 20 5 97

15 of 20 students handed in the extra problem with the results shown. Therefore the sum of the mean problem grades does not equal the mean total grade. Grades were slightly increased by the opportunity to do the bonus problem.

Unsurprisingly, the hardest problem to grade was problem #4 about Lagrange multipliers. I was primarily interested in checking the student's logical methods for solving this problem, and I was generous in giving partial credit for lapses in computation. Please note that the instructor would have lost 1 point for his original solution of this problem using the grading outline that follows! I came up with the following scheme to rationalize my grading of this problem only after several attempts:

PointsExplanation
4 These were setup points. Ideally I wanted to see the system of Lagrange multiplier equations: five equations in five unknowns. I interpreted this requirement generously, so even if no explicit statement of this system was given, I looked for evidence that all of the equations were used in the student's solution of the problem.
1 Consideration of the possibility that lambda could be 0. A student could earn this point also if I could conclude from work shown that only non-zero lambda's could occur in the solution of the problem.
2 Considering the possibility that z=w=0.
2 Deducing the two candidate critical points which result from z=w=0.
2 Stating that w=+/-z. Again, I would give this if I could deduce that students used this equation.
3 Obtaining the other 4 critical points of the problem. I deducted 1 point for reporting 2 or 8 critical points if all of the other work was correct. This is the point I would have lost: I also get careless about +/- signs.
2 Testing the objective function's values at the student's critical points (unless the student has radically simplified the problem), and then reporting the largest value as the max and the smallest value as the minimum.

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 final exam

Problem #1 Problem #2 Problem #3 Problem #4 Problem #5 Problem #6 Problem #7 Problem #8 Problem #9 Problem #10 Problem #11 Problem #12 Total 18 20 20 12 10 20 20 15 14 15 15 20 192 12 3 2 4 3 0 6 0 10 9 5 2 93 17.05 13.15 15.15 10.6 4.7 10.4 16.05 11.15 12.1 12.8 11.3 16.2 152.65 17.5 14 17 12 3 10 19 13 12.5 13.5 12.5 20 156

The course is supposed to be an honors calculus course. Perhaps the course t-shirt should not declare, "I survived Math 291", but, rather, "I survived the Math 291 final exam." As I graded the exam, after watching students whom I've known all semester work hard, I thought that the exam was perhaps a bit longer and a bit more difficult than what should have been given. We can discuss over a cup of tea some time exactly how large these two "bits" are!

Problems 6 (verifying Green's Theorem) and 11 (the intercepts of some tangent planes) were more difficult than I would give on a Math 251 final: more "abstract". Problem 10 (a racetrack with given curvature) asked for a serious conceptual rather than algebraic analysis. Problem 8 (asking for flux) was difficult, but I might have given it in a standard course without substantial expectation that correct solutions would be given. In fact, no one in the class answered the question totally correctly! The question can best be solved by using the Divergence Theorem on the whole boundary of a chunk of space -- you just can't forget about the bottom of the upper hemisphere! Perhaps the largest surprise on the exam was problem 5, asking for a second derivative of an implicitly defined function. That problem would certainly have been appropriate for the first exam in this course, and, actually, in most several variable calculus courses. I don't know why it was not answered correctly more often: just d/dx the equation twice, carefully of course, and try not to use any weird formulas! People also had some difficulty solving for the critical points in part a) of problem 2. Again, I was surprised. I do think there was some fatigue because this exam occurred at the very end of the exam period.

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]

Individual e-mail reports will be sent to students who requested this within 36 hours. It is now late afternoon on December 24.