The first exam

Numerical grades will be retained for use in computing the final letter grade in the course. Students with grades of D or F on this exam should be very concerned about their likely success in this course. Here are approximate letter grade assignments for this exam:

Discussion of the grading

Problem 1 (20 points)
a) 4 points for the correct derivatives, and 1 point for evaluating them correctly. I took off a point for evaluation errors, but then graded the remainder of the problem using the student's value of the gradient. I took off 1 point for each differentiation error.
b), c), d) f) e): a total of 15 points (5 points each) for the answers. I took off 1 point each if students "simplified" incorrectly.

Problem 2 (14 points)
a) 3 points for a correct formula. The phrase "functions of p and q" is unequivocal to me: a formula for r and s in terms of p and q. The "input" or domain variables are p and q, and the "output" or range variables are r and s. I took off 1 point for those students who did not declare which formula to use with which root, since they thereby impair communication consideratly later in the problem.
b) 1 point.
c) 5 points each for the r and s answers, allocated as follows:
1 point for each partial derivative, 1 point for each evaluation, and 1 point for assembling the change in the variable.

Problem 3 (16 points)
a) (8 points) 3 points for another vector in the direction of the plane, 3 points for the needed cross product, and 2 points for the answer.
b) (8 points) The student's answer to part a) may be used here. 3 points for putting the parametric equations into the plane equation, 2 points for simplifying and solving for the specific value of the parameter, and 3 points for inserting this value into the parametric equations and getting the answer.

Problem 4 (12 points)
2 points for z_s in terms of z_x and z_y. 2 points for z_t in terms of z_x and z_y. 4 points for (z_s)²+(z_y)² written as a multiple of (z_x)²+(z_y)². 4 more points for that multiple written suitably as a function of x and y.

Problem 5 (8 points)
3 points for one "straight line" limit. 3 points for another such limit not equal to the first. 2 points for the conclusion.

Problem 6 (12 points)
a) (6 points) I looked for evidence of two pieces, that each piece had a boundary showing (in space) y=x², and that each piece was planar.
b) (6 points) 2 points for reporting that f(x,y) is continuous at points (x,y) with y<x² and with y>x² and 2 points for some supporting reason. 1 point for remarking that f(x,y) was also continuous at (0,0) and (1,1) and 1 point for some supporting reason.

Problem 7 (10 points)
a) (4 points) 1 point for each derivative. 1 point for each evaluation.
b) (6 points) Here is how points were allocated using the strategy shown on the answers. Other strategies are possible, and I tried to grade them accordingly. 2 points for the needed scalar product; 2 points for the normal component; 2 points for the orthogonal component.

Problem 8 (20 points)
5 points for the graph of curvature and 3 points for the graph of torsion. I took 1 point off if either (or both) graph is not continuous. (Yes, we certainly can discuss this decision but it has been made.) The curvature graph should be 0 except for 2 positive bumps. The first, shorter bump should be about 3 units high, and the second, longer bump should be much less than 3. Otherwise the graph should be 0. The 0 behavior is worth 2 points, and each bump is worth 1 point. The difference in heights is worth 1 point. The torsion graph should be 0 except for one bump. The 0 behavior is worth 1 point, and the bump is worth 1 point. The longer bumps (the later one in the curvature case, and the only bump in the torsion case) should be flat, or else 1 point will be deducted in each case.

The second exam

Numerical grades will be retained for use in computing the final letter grade in the course. Students with grades of D or F on this exam should be very concerned about their likely success in this course. Here are approximate letter grade assignments for this exam:

Discussion of the grading

Problem 1 (15 points)
a) (6 points) I note that about half of the students who insisted on "simplifying" the answer made an error and lost a point.
b) (4 points) Just the picture.
c) (5 points) The most "economical" answer is what's on the answer sheet. I tried to give full credit to other answers which were also correct.

Problem 2 (12 points)
a) (2 points)
b) (3 points) I gave some credit here for a transformation which seemed to have some chance of working out!
c) (2 points) A picture.
d) (3 points) A computation.
e) (5 points) The whole computation, with the correct answer.

Problem 3 (12 points)
a) (4 points) I deducted 1 point for anything which made subsequent computation much easier.
b) (4 points) Almost any sketch which appeared rudimentarily correct will earn full credit. Things that show something horribly wrong won't.
c) (4 points) Compute the integrals. Those iterated integrals which were not similary in difficulty to the "correct" answer could not earn full credit.

Problem 4 (14 points)
a) (6 points) Don't forget the Jacobian!
b) (8 points) The answer with s and A was worth 4 points. Describing what happened as s-->0⁺ and the limiting value earned 3 points. I reserved 1 point for the special situation involving log and this point was lost if log wasn't mentioned.

Problem 5 (12 points)
I took off some points if the iterated integral had the wrong limits. Limits that did not have variables when they should have lost 2 points (!), as did limits in the theta variable which were incorrect. Note that correct solutions with iterated integrals in dz&nbps;dr d(theta) order certainly could be correct, and could earn full credit.

Problem 6 (12 points)
a) (10 points) 5 points for locating the critical point, and 5 points for analysis using the Second Derivative Test. I gave 4 points for writing some sort of mess which might locate the critical point.
b) (6 points) 3 points for locating the critical points, and 3 points for some analysis leading to correct conclusions about the critical points. I wanted some reasoning here.

Problem 7 (16 points)
a) (8 points) I ultimately searched for the extreme values of the objective function, f.
b) (8 points) Again, I ultimately searched for the extreme values of the objective function, f.

The final 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 (20 points)
a) (16 points) 8 points for finding the two critical points. 4 points each for correctly using the Second Derivative Test to find their nature. If only one c.p. is found, or more than one (!) then 6 of the first 8 points are earned, and only 6 of the 8 points for the Second Derivative Test can be earned.
b) (4 points) 2 points for finding the critical points with the correct gradient. 2 points for explaining why these critical points are all minima.

Problem 2 (20 points)
a) (12 points) 4 points for finding the correct gradient and evaluating it. 4 more points for the unit vector and 4 points for the magnitude.
b) (4 points) Writing an equation for the plane..
c) (4 points) Writing equations (or a vector expression) for the line. 1 point for the line going through (1,2,3). 0 points if what's given is not a line.

Problem 3 (20 points)
a) (12 points) 4 points for F_x and 4 points for F_y and then 4 points for getting the change in z. Equivalent work gets equivalent scores (for example, if change in an f defined by the lect hand side of the equation is set to 0 and the three derivatives of F are found, etc.).
b) (8 points) For finding the indicated derivative.

Problem 4 (20 points)
2 points for stating conditions on P and Q (these were explicitly requested!). 18 points fo a recognizable proof. 9 points each for the P and Q parts. I will try to split the 9 ponits into 3 points for analysis of the double integral, 3 points for analysis of the line integral, and then 3 ponts for showing that they are equal.

Problem 5 (20 points)
a) (12 points) Some evidence of antidifferentiation and matching up the results.
b) (8 points) For computing the result.

Problem 7 (20 points)
a) (10 points) For the process and answer.
b) (10 points) For the answer (4 points) and some indication of how it was obtained.

Problem 8 (20 points)
Preparation (10 points)
2 points of the computation of the gradient of W. 1 point each for the values of the grdient at the five requested points. 3 points for the values of the function W at the three requested points.
The drawing (10 points)
1 point for each of the five vectors, drawn with its "tail" at the correct point and with approximately correct magnitude and direction. 1 point for the x-axis correctly identified as a level curve, and 2 points for each of the other two level curves, approximately correctly sketched.

Problem 9 (20 points)
a) (5 points) The computation and answer.
b) (5 points) Outward (1 point) normal (2 points) of length 1 (2 points).
c) (5 points) Computation of the dot product is 3 points, and putting it into the appropriate integral is worth 2 points.
d) (5 points) Citation of Stokes' Theorem (2 points) and use of c)'s result (3 points).

Problem 10 (20 points)
a) (5 points) Need some sort of limit statement not just something about "division by 0 is bad": the penalty here is 2 points.
b) (15 points) Conversion to an iterated integral (3 points); evaluation of the inner integral (2 points); the outer integral (2 points): this is 7 points. A statement of {con|di}vergence needs two restrictions because the inner and outer integrals each need one. This is 5 points (2 points for missing one of the inequalities). Misstating the inequalities as one or more equalities will be penalized 1 point. Analysis of the log case is 3 points (hey, this was discussed in the solution to a similar problem on the second exam!).

Maintained by greenfie@math.rutgers.edu and last modified 12/27/2006.