The first 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

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.

The second 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

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

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.

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.

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.