## Grading guidelines for the second exam

### Background

Students should realize when they write answers that the grader ALREADY knows the answers. Students should show that they know the answers, and, more importantly, know why what they write IS the answer: show the process. Some problems have little need for displayed process (can you guess the process hidden by "5x7 -3x4 +2 --> 35x6-12x3"?). So there is generally available computer software that can compute derivatives of complicated formulas because that process is basically easy. BUT: there isn't much software capable of analyzing complex situations using various reasoning techniques. Therefore the grader will be principally interested in seeing your methods of solution.

Arithmetic errors will be penalized in the following way: -1 for the first error, and -1 for any additional errors. But students will need to follow the consequences - that is, they aren't allowed to just change their minds in the middle of a problem if their arithmetic errors have led to a situation more difficult to analyze than the correct one would have been!

Simplification is unnecessary unless specifically requested. So an answer which is (sqrt{3}+7)2 can be left that way instead of writing 52+14*sqrt{3} or the approximation 76.2487. The decimal number given is an approximation, and if an exact answer is requested, the approximation may be penalized. Sometimes (as in this exam) exact values of certain functions are needed, such as in problems 1, 2, 3 and 7. The statements of the questions should be a guide to that.

Other methods than are given in the "official" answers may certainly be valid strategies for these problems. The answers presented are not supposed to represent the only correct method of solution. Valid solutions of any type will be graded in a manner similar to what is described below.

### Discussion of grading for each problem

1. (12 POINTS)
2 POINTS for differentiating the polynomial correctly. 3 POINTS for finding the critical numbers correctly. 4 POINTS for realizing that the extrema must occur at either the endponts or the critical number (of this, 2 POINTS for an explicit statement: "briefly explain" validated). 3 POINTS for giving the correct extreme values.

2. (12 POINTS)
a) 6 POINTS: 3 POINTS for the product rule and 2 POINTS for the chain rule. 1 POINT for correct evaluation.
b) 6 POINTS: 4 POINTS for differentiating correctly (2 POINTS off if the chain rule is not applied). Evaluating the derivative is 1 POINT, and 1 POINT for the correct inference about the behavior of Q from the sign of the derivative.

3. (16 POINTS)
2 POINTS for a picture (a picture need not be drawn -- if there is successful work in the problem, these points will be given), 1 POINT for the constraint, 1 POINT for the constraint solved for one variable, 2 POINTS for the objective function, 1 POINT for reducing the objective function to a function of one variable, 2 POINTS for differentiating the objective function correctly, 2 POINTS for finding out where the derivative is 0, 2 POINTS for explicitly substituting correctly and finding the two dimensions (and explaining which is which!), and, finally, 3 POINTS for some explanation of why the answer found provides a minimum area.

4. (20 POINTS)
a) 4 POINTS: 1 POINT for the domain. 1 POINT for where the function is 0. 2 POINTS each for the asymptotes.
b) 5 POINTS: 2 POINTS for the derivative computed correctly. 1 POINT for detecting the critical number. 1 POINT for identifying the critical number (as to rel max or min or neither) and 1 POINT for explaining why.
c) 6 POINTS: 4 POINTS for computing and simplifying the second derivative. 2 POINTS for correctly explaining why the statement is correct (1 POINT for each assertion).
d) 5 POINTS: 3 POINTS for the picture and 2 POINTS for the labels requested.

5. (12 POINTS)
a) 1 POINT for finding the surface area correctly.
b) 11 POINTS: 2 POINTS for the derivative computed and evaluated correctly. 3 POINTS for the linear approximation formula applied correctly. 1 POINT for using and converting percentages correctly (there are two such applications in the problem). 3 POINTS for an explanation of the discrepancy between the true answer and the approximating answer, and 2 POINTS for either a correct supporting second derivative computation or a correct supporting picture.

6. (14 POINTS)
7 POINTS for the graph (a correct picture must include the data about the seven given points and the various limits). 1 POINT each for the relative extreme and inflection points (a total of 4 POINTS). 1 POINT each for the equations of the vertical and horizontal asymptotes (a total of 3 POINTS).

7. (14 POINTS)
a) 7 POINTS: 5 POINTS for correct use of the chain and product rules. 2 POINTS for solving for the derivative correctly.
b) 7 POINTS: 2 POINTS for "plugging in" to the original equation to get one linear equation. 2 POINTS for "plugging in" to the derivative (with the correct derivative information) to get another linear equation. 3 POINTS for solving these equations correctly.

### Exam outcome

About 95 students took this exam. Several versions of this exam were given. Overall, the mean grade achieved was 60.71, the median was 60, the standard deviation was 18.57, and the grades ranged from 19 to 94. A grade below 45 on this exam was unsatisfactory, equivalent to a letter grade of D or F.
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