### 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]

### Discussion of the grading

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.