Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | Total |
---|---|---|---|---|---|---|---|---|---|

Max grade | 20 | 14 | 16 | 12 | 8 | 11 | 10 | 8 | 98 |

Min grade | 3 | 1 | 0 | 0 | 0 | 0 | 2 | 4 | 28 |

Mean grade | 17.15 | 10.93 | 12.07 | 10.56 | 5.70 | 6.30 | 7.81 | 7.07 | 77.59 |

Median grade | 19 | 13 | 15 | 12 | 8 | 6 | 9 | 8 | 87 |

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:

Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|

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

**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=x^{2}, 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<x^{2} and with y>x^{2} 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.

Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | Total |
---|---|---|---|---|---|---|---|---|

Max grade | 15 | 15 | 14 | 13 | 12 | 16 | 16 | 98 |

Min grade | 10 | 2 | 4 | 0 | 1 | 0 | 2 | 19 |

Mean grade | 14.33 | 12.29 | 10.67 | 10.19 | 9.63 | 8.89 | 14 | 80 |

Median grade | 15 | 15 | 11 | 11 | 11 | 8 | 15 | 85 |

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:

Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|

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

**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.

Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | #9 | #10 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|

Max grade | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 17 | 197 |

Min grade | 12 | 8 | 12 | 0 | 12 | 0 | 12 | 0 | 10 | 11 | 107 |

Mean grade | 17.38 | 16 | 17.54 | 12.96 | 18.85 | 18.15 | 18.73 | 15.73 | 18.04 | 15.15 | 168.54 |

Median grade | 18 | 17 | 20 | 18 | 20 | 20 | 20 | 18 | 20 | 16 | 176.5 |

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:

Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|

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

**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.
**