## Grading guidelines for the second exam

### Background

The background that students should have in mind when they write answers is the following: 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. While, for example, there is generally available computer software that can compute derivatives of complicated formulas, there's not much software that can analyze 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 allow to just change their minds in the middle of a problem if their arithmetic errors have led to a more difficult situation 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) values of certain functions are supposed to be "simplified", such as in problem 2. The statements of the questions should be a guide to that. In other questions, algebraic manipulation of expressions is almost essential to success (for example, problem 4c).

Other methods than are given in the "official" answers may be good strategies for these problems so the answers presented are not the only valid solutions. Any correct solutions will be graded in a manner similar to what is described below.

### Discussion of grading for each problem

1. (17 POINTS)
In each case, 1 POINT off for minor errors, and 2 POINTS off for misuse of the chain/product/quotient rules. The last includes incorrect cancellations or combinations in and out of functions.
a) 5 POINTS
b) 5 POINTS
c) 7 POINTS (2 POINTS for successfully solving the differentiated equation for the derivative).

2. (17 POINTS)
a) 2 POINTS: just the answers.
b) 2 POINTS: again, just the answers, but as required by the statement of the problem.
c) 4 POINTS: 2 POINTS for the correct derivative, and 2 POINTS for detecting the root of G'(x) (1 POINT reserved for correct statement of the answer).
d) 4 POINTS: 2 POINTS for the correct second derivative (based on the student's first derivative) , and 2 POINTS for detecting the root of G''(x) (1 POINT reserved for correct statement of the answer).
e) 5 POINTS: 2 POINTS for the picture, and 3 POINTS for each labelled feature (the inflection point, the relative minimum, and the horizontal asymptote).

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 S and H, and, finally, 3 POINTS for some explanation of why the answer found provides a minimum area. I think this is a problem from the textbook.

4. (22 POINTS)
a) 4 POINTS: 2 POINTS for the roots of F', and 1 POINT each for identifying where F'>0 and F'<0.
b) 4 POINTS: 1 POINT each for identifying each critical number (as to rel max or min or neither) and 1 POINT each for telling why. 0 POINTS for just giving the critical numbers which would duplicate what was earned in a).
c) 5 POINTS: 2 POINTS for a correct expression for F'' and 1 POINT each for roots of F''.
d) 3 POINTS: 1 POINT each for indentifying each of the two points of inflection, and 1 POINT for telling why they are points of inflection.
e) 6 POINTS: 3 POINTS for the picture, and 3 POINTS for identifying the two points of inflection and the relative minimum. Drawing this graph to show the essential features is not easy.

5. (15 POINTS)
7 POINTS for getting an area formula (2 POINTS of these for declaring a correct area formula for a triangle). 3 POINTS for finding the first derivative and the critical number for the area formula. 2 POINTS for giving the vertices of the triangle. 3 points for saying why the triangle found has maximum area.

6. (13 POINTS)
a) 3 POINTS for giving the correct formula.
b) 10 POINTS: 1 POINT for finding theta at the specified time. 4 POINTS for correctly differentiating the formula in a) with respect to time. 1 POINT for finding (sec theta) squared at the specified time, and 2 POINTS for computing the right hand side at that time. 1 POINT for solving for the requested derivative correctly, and 1 POINT for declaring that it is decreasing.

### Exam outcome

About 95 students took this exam. Several versions of this exam were given, with statistics for the versions fairly close. Overall, the mean grade achieved was 53.8, the median was 54, the standard deviation was 20.1, and the grades ranged from 6 to 94. It should be noted that the number of students attending lectures for this class since the first exam ranged from a low of 53 to a high of 76, with the mean attendance approximately 67. In other words, about two-thirds of the students usually came to class. Although "the management" believes this was an appropriate, thorough and (in some aspects) challenging exam, many students who participated did not seem to take advantage of every opportunity to learn.