### The first exam

Problem#1 #2 #3 #4 #5 #6 #7 #8 Total
Max grade 10 12 14 20 8 8 18 10 96
Min grade 0 2 0 5 0 0 0 0 20
Mean grade 6.35 9.40 10.55 14.43 2.56 1.94 11.10 5.94 62.27
Median grade 8 10 11.5 15 2 1 11.5 6 61

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:

 Letterequivalent Range A B+ B C+ C D F [85,100] [80,84] [70,79] [65,69] [55,64] [50,54] [0,49]

An answer sheet for version A of the exam (with a green cover page) is available. The questions of version B were close to those of version A. I hope that students themselves will be able to create version B answers after reading the version A answers. 98 students took the exam. Statistical measures of the performance of the 47 students who took version A and the 51 students who took version B were quite close. The instructor was most surprised by the scores of problems 5 and 6.

Problem 1 (10 points)
2 points for correctly substituting this f(x) in the definition, including f(x+h). 2 points for correctly combining the fractions. 2 points for factoring out and canceling the h's. 2 points for taking the limit, and 2 points for recognizing the derivative.
0 points for a correct answer which is not supported by algebra. 0 points for quoting the definition of derivative, since the definition is given on the formula sheet. 2 points off for reporting that the derivative is a formula with two variables. This seems to have occurred because a limit was not taken.
The answer can be checked using the quotient rule.

Problem 2 (12 points)
a) 8 points. 2 points for getting a point (two coordinates) the line must go through. 2 points for computing the derivative. 2 points for getting a correct value of the derivative. 2 points for assembling and reporting a correct equation for the tangent line.
b) 4 points. 2 points for a rudimentarily acceptable curve (up and down, and through the origin). 2 points for a tangent line at the correct point.
Calculators and consistency
A graphing calculator is required for this course.
Therefore you should have such a device with you, and the curve sketch requested in this problem can be graded strictly. But this also means that you should have been able to compare the graph of the curve and the answer obtained in a), and the consistency of the two (picture and the tangent line equation) can be checked. For example, even a rough check should suggest a positive slope for the tangent line.

Problem 3 (14 points)
a) 8 points: 1 point each for the correct values of A and B, and 6 points for discussing how/why: some reasoning must be given. In particular, there must be explicit reference to either left/right limits at 0 and 1 and/or explicit consideration of the "other" functions, 3-x2 and 2x. Since we have studied continuity, the correct words and techniques (involving limits) are available.
b) 6 points: the graph should be continuous (!) otherwise -2 points. 2 points for the correctly drawn parabolic curve segment, 2 points for the correctly drawn line segment, and 2 points for the correctly drawn exponential curve segment. Each curve segment should connect the correct two end points (1 point) and should bend in the correct way (1 point): the parabolic curve should bend down, the straight line should not bend, and the exponential curve should bend up. Again, availability of a graphing calculator means that this aspect of the problem can be graded rather strictly.
2 points will be deduced for drawing something which is not the graph of a function (that is, several curves over the same x).

Problem 4 (20 points)
This problem was graded quite generously, perhaps to make up for the strict grading of the exam's graphical problems. Acceptable "evidence" was sometimes rather minimal.
Each part is worth 5 points: the answer alone is worth 1 point, and other work (how/why/explanation) is worth 4 points. Graphical information was acceptable verification on part b); alternatively, some comment on the behavior of the function must be given. In part d), you should know the value of cos([even integer]Pi) (1 point).

Problem 5 (8 points)
Parts a) and b) together are worth 5 points. Observing that 25 is positive and (-2)5 is negative is worth 2 of the 5 points. To earn the remainder, some idea should be given about the size (+/-) of the trig part of the function compared to the monomial.
c) 3 points: a citation of continuity and/or the Implicit Function Theorem should be given (1 point) along with a specific interval as answer.
Please note that verifying the conclusions of the problem using a specific value of K does not solve the problem written on the exam.
The instructor's error Version B changes the cosine to sine, which makes part c) rather simple: for any value of K, f(0) will be 0. Three students who took version B earned points with this observation.

Problem 6 (8 points)
The restrictions in domain obtained by considering the top are worth 5 points. The restriction from the bottom are worth the other 3 points when correct conclusions are obtained.
Since the instructions specified "Explain your answer algebraically" a graph cannot support the explanation, but a graphing calculator probably can be used to get hints.

Problem 7 (18 points)
2 points for each correct answer. Parts e) and g) are therefore worth 4 points each.
If an answer is an interval, I'll give 1 point for each correct endpoint. I'll be a bit sloppy and not worry about whether the endpoints themselves are in the intervals. When considering points of continuity and differentiability, I will deduct 1 point if any endpoints of the domain are included. The endpoints are not part of the domain and therefore are not eligible for continuity or differentiability of f(x).
Consistent with this, if the answer to a part of the problem is one or two values of x, I'll deduct a point for each extra (incorrect!) number you supply. The deduction will be limited by a score of 0 for the part!

Problem 8 (10 points)
a) and b) are worth 3 points each. Small errors (+/- signs, for example) will be penalized 1 point, while errors in the product or quotient rule will receive 2 point deductions. "Bald" answers with no supporting work will receive full credit here.
c) 4 points. 1 point for the answer, and 3 points for the process, roughly 1 point for each possible invocation of the product rule (so the first derivative can earn 1 point, and the second derivative can earn 2 points). I must see explicit evidence (with g(x), g´(x) and g´´(x)) that the product rule has been correctly applied.

### The second exam

Problem#1 #2 #3 #4 #5 #6 #7 #8 #9 Total
Max grade 15 12 10 12 8 8 17 9 8 86
Min grade 4 0 0 0 0 0 0 0 0 14
Mean grade 12.51 4.16 6.03 4.48 5.76 1.73 6.85 3.07 3.76 48.35
Median grade 14 2 6 4 7 1 6 2 4 50

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:

 Letterequivalent Range A B+ B C+ C D F [80,100] [75,79] [65,74] [60,64] [50,59] [45,49] [0,44]

An answer sheet for version A of the exam (with a blue cover page) is available. The questions of version B were close to those of version A (or identical!). I hope that students themselves will be able to create version B answers after reading the version A answers. 93 students took the exam.
Statistical measures of the performance of the 46 students who took version A and the 47 students who took version B were close. The overall low grades were not anticipated by the instructor.

Problem 1 (15 points)
a) 6 points. 1 point for the derivative, 2 points for the critical numbers, and 3 points for the intervals.
b) 4 points. 1 point for the (second) derivative, 1 point for the inflection point, and 2 points for the intervals.
c) 5 points. 2 points for sketching a continuous function which wiggles up and down in a reasonably correct manner; 3 points for the requested labels. Since students are assumed to have a graphing calculator and the axes given were very suggestively scaled axes, 1 point will be deducted if the graph egregiously does not fill the indicated window correctly. Identifying the ends of the graph as relative extrema is incorrect and will earn a 1 point penalty.

Problem 2 (12 points)
4 points for each asymptote. There are two horizontal asymptotes (+ infinity and -infinity) and one vertical asymptote. Of the 4 points, 2 are earned for the correct answer, and 2 for some justification. "Justification" will be interpreted very generously. 2 points are lost once in the problem for giving an asymptote only as a constant, since the equations of the asymptotes are requested. 1 point is lost once in the problem for giving a numerical approximation since the question states that "approximations are not acceptable."

Problem 3 (10 points)
2 points for a correct equation relating D and L and W.
1 point for finding D at "the certain time."
3 points for differentiating the equation connecting D and L and W.
3 points for substituting correctly in the differentiated equation and finding "how fast is the length of the rectangle's diagonal" is changing.
1 point for the answer, "decreasing."
If the problem has been simplified too much by an incorrect and much simpler equation relating D and L and W, only 3 points can be awarded of the 6 available for differentiation and substitution.

Problem 4 (12 points)
a) 2 points. Evidence should be given, or else the correct answer (both points needed!) earns only 1 point.
b) 10 points. 5 points for computing the derivative. 4 points for further computation (2 for symbolic work and 2 for numerical work) and 1 point for the answer.

Problem 5 (8 points)
a) 1 point.
b) 2 points for the derivative and 1 point for evaluation.
c) 2 points for the formula and 2 points for the evaluation.

Problem 6 (8 points)
Comment This is a composition and is not multiplication. Students treating this as multiplication will not receive any credit.
1 point for correct evaluation of the function value.
2 points for a correct formula for the first derivative and 1 point for the value of the first derivative.
3 points for a correct formula for the second derivative and 1 point for the value of the second derivative.

Problem 7 (18 points)
a) 2 points. 1 point for the answer, and 1 point for a supporting reason.
b) 3 points. 2 points for the answers (1 for each limit), and 1 point for supporting reasoning.
c) 9 points. 2 points for computing the derivative, and 3 points for manipulating the derivative so that information can be obtained. 2 points for the answers (1 point for each extremum) and 2 points for supporting reasoning.
d) 4 points. 2 points for the answer, and 2 points for supporting reasoning, which should refer both to results of b) and c). It is not sufficient to use the evidence from c), because a function with a relative max and relative min may well have a larger domain. One very simple example is given by the function in problem 1: look at its graph and compare the graph of the function in this problem!

Problem 8 (9 points)
3 points for the setup: constraint and objective function. 3 points for finding critical numbers. 3 points for checking values at endpoints and the interior critical number.
Comment I thought the phrasing "the product of the square of one multiplied by the other" was unequivocal, but some students seemed to misunderstand. I regret this. Perhaps I should have written, "the product of one of these numbers multiplied by the square of the other number." Improvement is good.

Problem 9 (8 points)
a) 4 points. Compute the derivative.
b) 4 points. Declare that f´(x)>0 appropriately, so that f(x) is increasing.

### The final exam and course grades

The final exam was graded by the entire instructional staff of Math 135. Approximately 700 students took the exam. The median grade was about 132. The raw scores, which a possible range of 0 to 200, were converted to letter grades as follows:

 Letterequivalent Range A B+ B C+ C D F [175,200] [165,174] [150,164] [135,149] [110,134] [100,109] [0,99]

I looked at the grading of each exam and then checked the addition on each exam. I corrected grading and addition errors (there were a few, but really not very many, and none of significance).

University rules state that I must retain possession of the exams but that you may certainly have access to them. If you wish to look at your final exam, please get in touch with me to find a mutually agreeable time. E-mail (to greenfie@math.rutgers.edu) is probably best.