The first exam

The two versions of the exam had very similar statistical results. 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 (12 points)
a) (6 points) A Quotient Rule error loses 2 points. More minor errors lose a point.
b) (2 points) 1 point for each answer. Knowing that e⁰=1 is worth 1 point in this problem.
c) (2 points) For any correct equation of the line.
d) (2 points) Geometric slope and point are each worth 1 point: the line should look tangent.

Problem 2 (8 points)
1 point for g(1); 2 points for g´(x) and 1 point for for g´(1); 3 points for g´´(x) and 1 point for for g´´(1). 2 points off for an error in the Product Rule or the Chain Rule.

Problem 3 (10 points)
2 points for correctly instantiating [f(x+h)-f(x)]/h. This is mostly for substitution of x+h into f (or something equivalent). If this formula is not quoted correctly, then no credit can be earned in this problem (it is difficult to compute f´(x) from the definition when the definition isn't known!). If somehow the student writes 3–x+h instead of 3–(x+h) the student loses these 2 points, but they can be the only 2 points lost of the 10 if everything else is correctly done.
3 points for correctly combining the fractions.
2 points for cancelling the h's.
1 point for the answer, which must be the correct answer for the student's work.
2 points off if there is no mention of lim_h→0 in an otherwise correct solution.
The answer alone or the answer obtained algorithmically earns no credit.

Problem 4 (12 points) Each part is worth 4 points. Simple arithmetic errors lose 1 point each. Errors with the Product Rule or the Quotient Rule lose 2 points each.

Problem 5 (12 points)
The idea of taking the derivative and trying to find x so the derivative is 0 earns 3 points alone (if this can be read sufficiently clearly).
The correct derivative is worth 5 points. Solving for an x where f´(x)=0 earns the remaining 4 points.
Credit can be lost for assuming incorrect algebraic properties of logs or exponents.

Problem 6 (12 points)
4 points for the hypotenuse in terms of x (2 of these for the "legs" of the right triangle, and 2 of these for correctly using this information to get the hypotenuse).
3 points for the correct total length of the horizontal and vertical sides.
2 points for a correct formula for f(x).
3 points for a correct domain (with or without either or both endpoints). 1 of these points is earned for a specification of x>0 or x≥0.

Problem 7 (8 points)
The word continuous in relevant context earns 2 points. The phrase Intermediate Value Theorem in relevant context earns 2 points. This means that just scrawling (scrawl: "write in a hurried untidy way") the word "continuous" or the phrase "Intermediate Value Theorem" will not necessarily earn credit. 1 point is earned for a function value which is positive, and 1 point is earned for a function value which is negative. Finally, an explicit correct answer supported by the student's work earns 2 points (an interval of the form [0,something] in a similar context earns 1 point).
Comment The problem statement contains the following:

You must give evidence supporting your assertion -- the interval given can be any valid interval. The evidence should include specific vocabulary and results from this course.

Problem 8 (12 points)
a) (5 points) Factoring the top and bottom is worth 3 points. Cancellation and evaluation earns the remaining 2 points, 1 of which is for the answer.
b) (2 points) A correct answer earns 1 point. Some support (the word "continuous" or the phrase "plugging in") earns a second point.
c) (5 points) A correct answer earns 1 point. Some analysis (or even just acknowledgment!) of absolute value earns 2 points, and then algebra using this gets 2 more points.

Problem 9 (14 points)
a) (7 points) Restricting attention to 0 and 2 is worth 1 point. 2 points for getting a constraint at 0. 2 points for a similar constraint at 2. 2 points for solving and reporting correct values of A and B.
b) (7 points) The graph should be continuous (2 points). 2 points for the line segments (each correct line segment earns 1 point). 3 points for an increasing, correctly curved parabolic arc (not curved: 2 points off).

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

Special note
I became convinced after beginning the grading that the work required for problems 7 and 8 was not equivalent so awarding equal points (10 each) was not appropriate. I changed the point value for problem 7 to 9 points and for problem 8 to 11 points. The following credit reflects this decision.

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 (8 points)
Differentiation of the equation is worth 5 of the 8 points. Each Chain Rule error loses 2 of these points. 3 points are earned for correctly solving for y´. Students who simplify the differentiated equation with a Chain Rule error are eligible for only 2 of the 3 "solving" points.

Problem 2 (10 points)
a) (5 points) 1 point for the derivative of arctan, 2 points for correctly using the Chain Rule, and 2 points for correctly using the Quotient Rule.
b) (5 points) 3 points for the derivative, and 2 points for an equation of the tangent line (1 point for a correct point and 1 point for a correct slope). Please note that if the derivative is not evaluated to get a number, the graph of the resulting equation is not a line, so it is eligible for only 1 of the 2 equation points.

Problem 3 (12 points)
4 points for a correct equation connecting the distances at a "general" time, 4 points for correctly differentiating this equation, and then 4 points for correctly instantiating the equation with specific information and deducing the desired rate of change.

Problem 4 (6 points)
1 point for correctly selecting a function to use, 1 point for the derivative, 2 points for the values of the function and the derivative, and then 2 points for answering the question (correct assembly of all of the information in something related to the linear approximation).

Problem 5 (8 points)
Numerical estimates for x₁ and x₂ will earn 2 points each. The estimates need not be identical to those given in the instructor's solution, but certainly x₁ should be larger than 2 -- in fact, close to 3, and x₂ should be between 2 and x₁, and close to 2. Each line segment is worth 2 points. They should be tangent to the curve drawn (starting reasonably close to 1) and tangent at the correct points.

Problem 6 (10 points)
A statement or use of the idea "Extreme values occur at {endpoints & critical points}" earns 1 point. Finding the values at the endpoints earns 2 points. A correct derivative earns 2 points, and finding the critical point earns 2 points. Finding the value at the critical point earns 1 point. Finally, 2 points are awarded for the correct answers.

Problem 7 (9 points Please see the note preceding these remarks.)
The correct derivative earns 2 points. Finding the correct two critical points earns 2 points. Finding the correct intervals of {in|de}crease earns 1 point each (some reasoning must be supplied for each interval) for a total of 3 points. Applying the First Derivative Test and making the appropriate conclusion for each critical point earns 1 point each, for a total of 2 points. Lose 1 point to problem 8.

Problem 8 (11 points Please see the note preceding these remarks.)
2 points for the first derivative and 3 points for the second derivative. 3 points for finding the candidates for inflection points (1 of these is for writing f´´(x)=0). 1 point for each interval & concavity diagnosis for a total of 3 points.

Problem 9 (4 points)
Each picture earns 2 points, with 1 of the 2 earned for the correct {in|de}creasing behavior and the other for correct concavity.

Problem 10 (12 points)
Setting up the problem earns 3 points: 1 point for area (constraint) and 2 points for the cost (objective function). No points will be earned if perimeter is used instead of cost. Students who consider an incorrect objective function will still be eligible to earn 8 of the 12 points for this problem if their objective function is as complicated algebraically as the correct objective function. They can't get the points for the objective function or for the answer.
Then 3 points for changing to a 1 variable problem and 2 points for finding the critical value and 2 points for explaining why a minimum has been found. Some methods different from the supplied answer are certainly valid: students could analyze the first derivative ("First Derivative Test") or they could use the second derivative and conclude that a minimum has been found using concavity.
2 points for the answer (both dimensions are requested!).

Problem 11 (10 points)
Algebraically combining the pieces correctly earns 3 points. The pieces cannot be analyzed separately! A first use of L'H is worth 3 points: 1 point for considering eligibility, and 2 points for the differentiation. A second use of L'H is worth 3 points: 1 point for considering eligibility, and 2 points for the differentiation. The final answer is worth 1 point.
This is a difficult problem.

The final exam and course grades

Let's see: there were several arithmetic errors and a number of 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 there were a handful (fewer than 5 changes) which were larger. I'm glad I reread the exams but I don't think that I found and made any changes which were material (that has not been the case with other courses). Here are some statistics about the final exam.

The performance of 153 was, on the whole, somewhat better than might have been predicted statistically for groups of students with similar preparation in other instantiations of Math 151. I think a number of students prepared very diligently for the final exam and these numbers were useful in reporting course grades (more details on that are below). 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. Many students made errors involving high school algebra. I think the final exam was generally straightforward and most 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 153, 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 the peer mentors), workshop and quiz grades (reported by Mr. Nakamura), and the number of QotD answered by students.
I computed a number for each student essentially weighted as previously described. Attendance was generally very good (thank you!) and I only thought of penalizing 3 students for poor attendance (their own efforts effectively penalized them!). Therefore the student numbers were constructed from the exams, the textbook homework, the number of QotD answered, the workshop grades and quiz 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 Monday, December 21. 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.
Students with grades of D should be especially concerned about their preparation and likely success in the second semester (Math 152) of the calculus sequence. Although students can continue with this grade, experience and statistics suggest that the chance of future success (passing that course!) is small. If you do decide to continue, please realize that you will need to allow extra effort and extra time, and that you'll need to do a great deal of dull, repetitive practice and serious study.

Maintained by greenfie@math.rutgers.edu and last modified 12/21/2009.