Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | Total |
---|---|---|---|---|---|---|---|---|---|
Max grade | 16 | 12 | 10 | 15 | 12 | 10 | 12 | 12 | 97 | Min grade | 2 | 0 | 0 | 1 | 0 | 4 | 5 | 2 | 52 | Mean grade | 13.44 | 7.8 | 4.22 | 11.56 | 8.2 | 9.08 | 9.68 | 7.68 | 74.16 | Median grade | 15 | 8 | 7 | 12 | 9 | 10 | 10 | 8 | 75 |
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 | [85,100] | [75,84] | [70,74] | [65,69] | [55,64] | [50,54] | [0,49] |
Grading guidelines
Minor errors (such as a missing factor in a final answer, sign error,
etc.) will be penalized minimally. Students whose errors materially
simplify the problem will not be eligible for most of the
problem's credit.
The grader will read only what is written and not attempt to guess or
read the mind of the student.
The student should solve the problem given and should not invent
another problem and request credit for working on that problem.
Problem 1 (16 points)
a) (2 points) The region should be roughly three-sided with line
segments on the x- and y-axes and with a curvy concave down boundary
curve for the third side. If no label is present, any positive score
will be reduced by 1 point (a label was explicitly requested).
b) (4 points) Setting up the integral is worth 2 points, and
evaluating it with C correctly used is worth the other 2 points. 1
point will be given for an indefinite integral.
c) (8 points) Setting up the integral either dx or dy is worth 5
points. Computing it is worth the other 3 points. 3 points will be
given for an indefinite integral.
d) (2 points) Setting the answer in c) equal to 1 earns 1 point and
then solving the equation earns 1 point.
Problem 2 (12 points)
a) (4 points) 2 points for the substitution and 2 points
for antidifferentiation.
b) (8 points) 2 points for an initial substitution, then 5 points for
an integration by parts which "moves forward" and then 1 point for the
final answer. Integration by parts with the original variable is
certainly also valid, and will be scored appropriately. An integration
by parts which "pumps up" the powers earns 2 points only.
Problem 3 (10 points)
a) (3 points) 1 point for the largest value and 1 point for a reason,
and then 1 point for the limit answer.
b) (5 points) 2 points for setting up the mean or average value
integral correctly, 2 points for the antidifferentiation, and 1 point
for the limit answer.
c) (2 points) 1 point for the answer and 1 point for some explanation.
Problem 4 (15 points)
Proportional sides equation/information is worth 3 points. Density
information correctly used is worth 3 points. The distance to lift the
slice is worth 3 points. If either the density or the distance to lift
the slice is missing, the maximum which can be earned is 10 points. If
both are missing, the maximum which can be earned is 7 points.
Assembling this into a correct definite integral is worth 3 points
(essentially the student is penalized 2 of these 3 points if one of
density/distance is missing). The final computation is worth 3
points. A missing factor of Π loses a point.
Problem 5 (12 points)
3 points for a correct trig substitution, 3 more points for conversion
of the integral correctly into the trig "variable", 2 points for a
correct antidifferentiation, 2 points for converting correctly back to x, and 2 points for the answer.
Problem 6 (10 points)
2 points for writing the correct symbolic sum, and 2 points for
combining terms successfully. 1 point each for correct values of the
variables for a total of 3 points. 2 more points for correct
antidifferentiation, 1 for each term.
Problem 7 (12 points)
4 points for the (cos(x))2 antidifferentiation, 4 points for the
(cos(x))3
antidifferentiation, 2 points for correctly combining them, and 2 more
points for the final answer, including evaluating the trig functions
correctly.
Problem 8 (13 points)
a) (4 points) Writing the correct weights earns 1 point, getting
Δx correct earns 1 point, and using the correct function values
earns 2 points.
b) (7 points) Computing the first derivative earns 1 point, and
computing the second derivative earns 2 points. Estimation is worth 4
points.
c) (2 points) Setting up the inequality correctly is worth 1
point. Using it to get useful information about a correct N earns the
other point. If the inequality is incorrect, no points are earned in
this part. An equality with N loses 1 point.
Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | #9 | #10 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|
Max grade | 10 | 10 | 9 | 8 | 10 | 8 | 10 | 12 | 6 | 9 | 80 | Min grade | 1 | 1 | 0 | 1 | 0 | 6 | 2 | 0 | 0 | 0 | 45 | Mean grade | 6.96 | 7.64 | 5.76 | 5.96 | 7.76 | 7.4 | 9.4 | 6.64 | 2 | 2.76 | 62.28 | Median grade | 9 | 8 | 6 | 8 | 10 | 8 | 10 | 6 | 1 | 2 | 64 |
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 | [80,100] | [70,79] | [65,69] | [60,64] | [50,59] | [45,49] | [0,44] |
Grading guidelines
Minor errors (such as a missing factor in a final answer, sign error,
etc.) will be penalized minimally. Students whose errors materially
simplify the problem will not be eligible for most of the
problem's credit.
The grader will read only what is written and not attempt to guess or
read the mind of the student.
The student should solve the problem given and should not invent
another problem and request credit for working on that problem.
Problem 1 (10 points)
1 point for y´ and 2 points for a correct initial instantiation
of the integral for surface area. Then 3 points for algebraic
transformation of the integrand. 2 points for antidifferentiation, and
2 points for work leading to the answer which was given.
Problem 2 (10 points)
4 points for the antidifferentiation of xe-mx; 2 points for
a substitution from 0 to a parameter; 2 points for use of L'H with 1
additional point earned for citing the eligibility of the quotient; 1
final point for getting the given answer.
Problem 3 (5 points)
a) (5 points) Taking ln's earns 1 point; algebraic reassembly as a
quotient earns 1 point; 1 point for recognizing eligibility for L'H; 1
point for use of L'H; 1 point for the final answer (using
exponentiation).
b) (5 points) Taking limits in the recursion is worth 2
points. Citation of the necessity that the limit (if it were to
exist!) is >0 is 1 point (or coping with this in some other
fashion); 2 points for the conclusion that the limit does not
exist.
Problem 4 (8 points)
4 points for the answer and 4 points for some supporting evidence.
Problem 5 (10 points)
A correct inequality comparing terms of the given series with a
simpler series is worth 3 points. A citation of the correct infinite
tail of the comparison series is worth 2 points, and summing that
infinite tail is worth 3 points. Final selection of a correct N is
worth 2 points.
Problem 6 (8 points)
Writing the integrals is worth 2 points. Computing the antiderivatives
is 2 more points, with 1 point for the "+C". 1 point is earned for
using the initial condition and 2 points for solving for y as a
function of of x.
Problem 7 (10 points)
6 points for the slope field elements. 2 points of that is reserved
for the 9 horizontal elements, 2 points for the positively sloped
elements, and 2 points for the negatively sloped elements. The
different slopes should be approximately correct (that's 1 of the 2
points for the +/- elements).
2 points each for the limit answers: 1 point for ∞ and 1 point
for the correct sign.
Problem 8 (12 points)
a) (3 points) For the answer (1 point for each term).
b) (9 points) Use of a graph to get an appropriate constant K is worth
2 points. Then applying it in a Taylor error estimate gets 2
points. Showing that the desired inequality is true is worth 3
points. The actual polynomial instantiated satisfactorily is worth the
final 2 points (factorials count for 1 of these 2).
Problem 9 (10 points)
Certainly there are several different approaches possible (two are
indicated in the answers distributed). Initiating an approach which
can be successful is worth 3 points, and then carrying it out is 5
points. The last 2 points are given for the answer.
Problem 10 (12 points)
a) (6 points) The θ part of the description is worth 2 points
and the r part is worth 4 points.
b) (6 points) Correct instantiation of the area integral is 2 points,
then recognition that it can be antidifferentiate to get tan is 2
points. Correct computation of the answer is worth 2 points. The
computation should use integration in polar computation.
The final exam and course grades
Grading the final exam
The final exam was graded by the instructors of Math 152 on Thursday,
December 17 following a previously written "rubric" or grading
guide. I have a copy of this guide, and I then sat and reread the
exams and the grades given for about 4 or 5 hours on Friday, December
18.
Let's see: there were several arithmetic errors and a host (well, about 10) situations where the grades given seemed to be different from the rubric's suggestions. Almost all of the grading changes I made were +/– 1 or 2 or 3 points, but in the case of one exam the change was more material, so I'm glad I reread the exams. Here are some statistics about the final exam.
Students | # of students | Grade range | Median grade |
---|---|---|---|
All Math 152 | 467 | 9 to 198 | 139 |
Math 152H | 25 | 105 to 193 | 170 |
The mean grade for 152H students was 165. I don't know the mean for the whole group of 152 students. The 152H students generally did well: congratulations to you. Here are the letter grade assignments for all students who took the exam.
Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|
Range | [180,200] | [167,179] | [154,166] | [135,153] | [117,134] | [110,116] | [0,109] |
Humans generally behave consistently. Some students lost points because they simplified incorrectly where this was not requested. Some students invented their own problems, and these may not have been related to what was asked. I think the final exam was generally straightforward and those students I asked told me that there was enough time to write and check solutions.
Use of these grades
The final exam letter grades are given to allow individual lecturers,
who are responsible for reporting course grades, to align the
performance of their specific groups of students with the overall
performance of students in the course. Random chance may give one
lecturer a group of students with better or worse performance than the
overall population, and, indeed, specific scheduling requirements may
force specific subpopulations with different math preparation to
enroll in some lectures. The common final exam and grading are part of
the faculty's effort to assign appropriate course grades, with equal
grades for equal achievement. In the case of 152H, the comparison of
final exam results of the section with the performance of "all
students" certainly helps me to assign grades which reflect student
performance in the "standard" sections.
Course grading
The information I had included the following: grades for the three
exams (two during the semester and the final), textbook homework
scores (reported by Mr. Conti), and workshop grades I assigned.
I computed a number for each student essentially weighted as previously described. I had
no quizzes, so that component was reduced to 0. I only asked 5 QotD's
which could serve as a proxy for attendance, but I decided that number
was too small to be helpful and discarded it. Anyway, attendance was
generally very good (thank you!). Therefore the student numbers were
constructed from the exams, the textbook homework, and the workshop
grades, with weights as described. I then assigned a tentative letter
grade based on breakpoints proportionately derived from the "bins"
shown above for the exams. I examined each student's record to make
sure that this process had not distorted or misrepresented student
achievement. I entered the course grades into the Registrar's computer
system on Sunday morning, December 20. I hope students will be able to
see them soon.
If you have questions ...
Rutgers requests that I retain the final exams. Students may ask to
look at their exams and check the exam grading. These students should
send me e-mail so that a mutually satisfactory meeting time can be
arranged. Students may also ask how their course grades were
determined using the process I described. Probably e-mail will be
sufficient to handle most such inquiries.
Maintained by greenfie@math.rutgers.edu and last modified 12/19/2009.