## 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, perhaps 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 in each problem 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 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. Numerical answers gotten from graphical information should be reasonably close. Thus in problem 7, the third critical number should be about 1.9, and 2 or 22 would not be reasonably close.

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 way. Valid solutions of any type will be graded in a manner similar to what is described below.

### Discussion of grading for each problem

1. 9 POINTS
5 POINTS for differentiating correctly, and then 2 POINTS for solving for the derivative correctly. 1 POINT off for minor errors, and 2 POINTS off for misuse of the chain or product rules. 2 POINTS for writing some correct equation for the tangent line.
Some students seem to have solved for y explicitly. So for this solution, the following point allocation is given: 2 POINTS for solving explicitly and correctly for y as a function of x. 5 POINTS for correct differentiation and correct evaluation. 2 POINTS for writing some correct equation for the tangent line.

2. 10 POINTS
2 POINTS for differentiating the polynomial correctly. 2 POINTS for finding the critical numbers. 4 POINTS for realizing the the max/min must occur either at the endpoints or the critical number. 2 POINTS for giving the correct extreme values.

3. 21 POINTS
a) 1 POINT for the correct root.
b) 3 POINTS: 2 POINTS for the derivative computed correctly and 1 POINT for the correct root of the derivative.
c) 3 POINTS: 2 POINTS for the second derivative computed correctly and 1 POINT for the correct root of the second derivative.
d) 14 POINTS: 6 POINTS for the algebraic answers distributed as follows: 1 POINT for the interval of increase; 1 POINT for the interval of decrease; 1 POINT for the interval which is concave up and 1 POINT for the interval which is concave down; 1 POINT for the correct pair of numbers; 1 POINT for the correct pair of numbers. 8 POINTS for the sketch: 1 POINT for each label (6 POINTS total) and 2 POINTS for the correct picture.

4. 10 POINTS
a) 3 POINTS for giving the correct formula.
b) 7 POINTS: 1 POINT for finding theta at the specified time. 3 POINTS for correctly differentiating the formula in a) with respect to time. 1 POINT for correctly finding (sec theta)2 squared at the specified time, 1 POINT for evaluating the right-hand side correctly, and 1 POINT for getting the final answer correctly.

5. 14 POINTS
4 POINTS for writing the correct area formula. 3 POINTS for differentiating it correctly. 2 POINTS for finding the critical number. 2 POINTS for finding the dimensions of the rectangle. 3 POINTS for saying why the rectangle is found has maximum area.

6. 12 POINTS
a) 3 POINTS for finding the derivative correctly. 3 POINTS for writing and using the approximation formula correctly.
b) 3 POINTS for an explanation, and 3 POINTS for a correct supporting computation.

7. 24 POINTS
a) 3 POINTS: 1 POINT for each critical number.
b) 6 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) 4 POINTS: 1 POINT for each interval correctly named.
d) 3 POINTS: 1 POINT for each inflection point.
e) 8 POINTS: 5 POINTS for labels (1 POINT each), and 3 POINTS for the curve.

### Exam outcome

About 90 students took this exam. Several versions of this exam were given, with statistics for the versions within 2 points of each other. Overall, the mean grade achieved was 48.8, the median was 48, the standard deviation was 20.0, and the grades ranged from 9 to 99. A grade below 45 on this exam was unsatisfactory, equivalent to a letter grade of D or F.