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

Problem #1
The definition was worth 2 points. Part b) was worth 10 points. Writing only a correct formula for f'(x) there earned nothing.

Problem #2
a) and c) were straightforward computations. In part b), the graph of the cubic earned points for showing the correct asymptotic information, the correct roots, and the correct approximate locations of "bumps". The line should have been drawn tangent to the curve.

Problem #3
Each part was worth 5 points. The correct answer earned a point. The other 4 points for each part were given for some correct justification. Numerical substitution alone was not considered sufficient justification.

Problem #4
This was the most difficult problem to grade (indeed, grading it took more time than grading 4 other problems!). A strong effort was made to grade fairly, with partial credit given for specific facts such as the computation of f(0) in part a). Statements about x² getting "very big" in part b) did not earn credit. The problem asks for very specific information about f(x) for x>=2, and so precise information (including estimates!) are needed. Please note that the input to sine will always be in radians in this course unless otherwise specified. Therefore the sign of sin(80) or sin(140) are not obvious without more discussion. (One is positive and one is negative, actually!) Successful arguments in both parts used estimates of sine's values: all that is needed is the range of sine, the interval [-1,1]. The use of the Intermediate Value Theorem in part a) needed citation of the necessary hypotheses, including continuity of the function and the difference in signs of the function's values at the endpoints. There are simple examples of discontinuous functions with no roots whose values are both positive and negative. The Intermediate Value Theorem is not relevant to part b).

Problem #5
Again, each part was worth 5 points. The lecturer missed a minus sign in one of these answers, so he would have gotten 99 on this exam.

Problem #6
Specific points were assigned to each feature of the graph of f'(x). Certainly there is room for honest disagreement since only qualitative information is given. For example, students received full credit if their graph of f'(x) appproached a finite limit (with the correct sign!) as x-->B⁺. The lecturer's answer indicates an asymptote on the graph of y=f'(x), but that does not seem to be forced by the graph of f(x). The vertical and horizontal asymptotes should have been written as equations: 1 point was deducted if they were not. Erroneously giving more or fewer points of discontinuity and non-differentiability was penalized.

Problem #7
Few students were successful in this problem, either because it was difficult or because it was the last problem on the exam. A few points of partial credit were given for a useful sketch or for writing the derivative of 1/x.

Overall results
Overall class performance was definitely weaker than the instructional staff anticipated and wanted. The lecturer wanted students to do "well" on problems 4 and 7. Most of the other problems are algorithmic and could be done by a computer program. The lecturer believes that problems 4 and 7 distinguish those students who can use the ideas of the course.

The second exam

Numerical grades (the Total) 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
a) was worth 4 points. No formula for linear approximation/linearization/differential received 0 points. f'(x) computed correctly was worth 1 point, and the correct value of f'(2) received another point. 2 points were given for applying the formula correctly.
b) Asserting that the function was concave down earned 1 point. 1 point was given for computing f''(x) and 1 point for computing f"(2) or f"(1.97). Giving a reason why the graph was concave down earned the fourth point.

Problem #2
a) 2 points for the drawing and 2 points for the limit of the sequence. In version B, we felt that the limit resulting from a₀ could be argued about: the exam writer wanted the limit to be +infinity, but some students drew convincing tangent lines and got the limit to be R₂. If there was a coherent picture and discussion, these students received full credit.
b) 2 points for the drawing and 2 points for the limit.
In each part, 1 point was given for having only the first point correctly drawn.

Problem #3
4 points if y' is correct. 1 point if the result is set equal to 1. 5 more points for finding the specific points on the ellipse which are requested. Partial credit was given for partial progress. 8 points were given if only one correct point was presented as the answer.

Problem #4
4 points for a correct first antiderivative (that is, f'(x), with +C) and 1 point for evaluating that constant using the initial condition. Another 4 points were given for a correct second antiderivative (f(x), with +C) and 1 last point for evaluating the second constant. Note that the first constant appears as part of the second antiderivative.

Problem #5
Correct statement of the constraint (an equation connecting V and r and h) earned 2 points. Correct statement of the objective function (the sum of the area of the bottom plus the area of the side) earned 3 points. Use of the constraint to obtain a function of one variable which must be minimized earned 3 points. Differentiation of this equation earned <3 points. Solving for the unique critical point earned 1 point and solving for the value of the other variable, 1 point. I gave 2 points for explaining why a minimum was found.
I do not apologize for any "abstraction" in this problem (that is, using "V" rather than 38, say). The problem is quoted directly from the text. Students in this course should be able to handle a some abstraction.

Problem #6
I found this problem difficult to grade, and tried to give partial credit carefully. I believe that the key to this problem is doing as little (!) work as possible by carefully using the information given. Since f'(x) is given in factored form, parts a) and b) should be easy. Please do note that x²-3 has two roots, not just one!
a) 4 points, one for each interval.
b) 3 points, one for each extreme value.
c) 1 point for correct computation of f''(x) and then 3 points, one for each interval of concavity.
d) 5 points. I tried to see if the graph either was correct or was consistent with the student's previous information. I looked for correctly labeled extreme points and inflection points.

Problem #7
10 points: 2 points for a general "area of a triangle formula", and 1 point for using the formula to compute the base of the triangle at the time given; 3 points for differentiating correctly; 2 points< for using the given rates of change correctly; finally, 2 points for obtaining the correct answer. Students can also do this problem by "solving" for the base as a function of area and height, and then using the quotient rule, and full credit was given for solutions of that type.
Students who wrote a blatantly incorrect version of, say, the product rule, were not given high grades on this problem.

Problem #8
Each part was worth 5 points. 1 point was reserved for the correct numerical answer, and the other points were earned by supporting evidence. As mentioned in class, calculator evidence alone is not valid. L'Hopital's rule is not relevant to part c) at all, and attempts to use it (usually characterized by writing some variant of the derivative) were given no credit.

Problem #9
8 points. From the first exam. I looked for some evidence that more was known. Similar computations have been needed in a number of applications since the first exam, so I hoped that additional "practice" would have resulted in more success here. This does not seem to be the case.

Overall results on the first 9 problems
This exam was probably too long. I don't believe any single problem was inappropriate or too difficult, but a better exam would have had only 8 of these problems rather than 9.

Grading the Bonus Problems
Problem 1 was graded mostly by glancing at the answer. If it was wrong, further investigation was needed.
In problem 2, various routes to the correct answer are possible, and a number of algebraic versions of the correct answer are possible. Therefore the process and answer were both checked. Some students gave only the first derviative which is not what was requested.
Few people were totally successful doing problem 3. 1 point was given for a correct algebraic statement of the constraint, and another point for correctly using the constraint to reduce the objective function from two variables to one. If the one variable objective function was correctly differentiated, a third point was earned.
Many people were successful doing the fourth problem. A few students, however, seemed to invent and try to answer their own problem(s), which usually seemed more difficult.
Problem 5 needed some explanation to score full credit. 1 point was given for the correct answer, and another point for differentiating a polynomial (or two). Finding the roots of these polynomials earned another point. But full credit was only awarded for making correct logical deductions from the evidence to support the correct answer.
Most students got problem 6 correct. Partial credit was given for small mistakes, but more points were taken off for serious misuse of the chain and/or product rules.
73 students gave answers to these bonus problems. The correlation coefficient for the bonus problem scores and the subtotals was .689, quite high (not surprisingly).

The final exam

The final exam itself
Current course policy is to keep the questions on the final exam confidential. Also, university regulations say that final exams must remain in the custody of the department. Students may examine their exams by making an appointment with the lecturer.

The grading of the final exam was straightforward. My personal opinion is that the exam was slightly easier than other recent Math 151 finals. I will write no more about the grading, except to declare that Ms. Calinescu and I have taken enough virtual airplane flights and searched diligently for ways to make functions continuous.

The median grade for the approximately 600 students who took the final exam was 135. The median grade for students taking the exam in sections 4, 5, and 6 was 148, and the mean for these students was 126.

Numerical grades (the Total) were retained for use in computing the final letter grade in the course. Here are letter grade assignments for this exam:

Maintained by greenfie@math.rutgers.edu and last modified 1/9/2003.