## Grading guidelines for the first 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 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. Sometimes (as in this exam) exact values of certain functions are needed, such as in problems 1, 2, 3 and 6. 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 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. (20 POINTS)
Each part is worth 5 POINTS: the answer alone is worth 1 POINT, and other work (how/why/explanation) is worth 4 POINTS. A graph can give acceptable verification for parts c) and d); alternatively, some comment on the behavior of the function must be given.

2. (14 POINTS)
a) 8 POINTS: 6 POINTS for discussing how/why: some reasoning must be given. 1 POINT each for the correct values of A and B. We've studied continuity, so 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 cosine curve (please note that -Pi/2 is between -2 and -1), and 2 POINTS for each correctly drawn line segment. -2 points for graphing several functions over the same interval.

3. (12 POINTS)
Each part is worth 6 POINTS: 4 POINTS for the formula, and 2 POINTS for the domain. The correct endpoints for the domain must be given, but either endpoint may be included/excluded from the domain because the physical setting of "A ladder ..." can really be seen in various ways.

4. (12 POINTS)
a) 8 POINTS: 2 POINTS for the statement of the definition of f´(x) (leaving out "lim" in the definition loses 1 POINT!), and 6 POINTS for successfully manipulating the difference quotient and getting the derivative. 0 POINTS for a correct answer which is not supported by algebra.
b) 4 POINTS: 1 POINT for getting the slope of the line, 1 POINT for getting the y-intercept or some point on the line, and 2 POINTS for giving a valid equation for the tangent line. -1 POINT for presenting an equation (such as (y-y0) DIVIDED by (x-x0) = m) which is not satisfied by EVERY point on the line!

5. (9 POINTS)
Each part is worth 3 POINTS. The answers alone can be written with little effort.

6. (8 POINTS)
2 POINTS for j(0), and 3 POINTS for each of j´(0) and j´´(0). The numerical values of each of these are worth 1 POINT, so if a value is omitted, -1 POINT.

7. (13 POINTS)
a) 2 POINTS: there are three places where Q is 0. -1 POINT each for missing any one of them (up to -2 POINTS, of course).
b) 4 POINTS: there are two intervals where Q is positive. 1 POINT for each correct end point of an interval, correctly presented. If an end point is included in an otherwise correct answer, -1 POINT (taken off only once in this section of the problem).
c) 3 POINTS: there are four places where Q´ is 0. -1 POINT each for missing any one of them (up to -3 POINTS, of course).
d) 4 POINTS: there are two intervals where Q´ is positive. 1 POINT for each correct end point of an interval, correctly presented. If an end point is included in an otherwise correct answer, -1 POINT (taken off only once in this section of the problem).

8. (12 POINTS)
3 POINTS for writing a correct equation using the two formulas. 3 POINTS for writing a correct equation using the derivatives. 6 POINTS for solving these equations, with 3 POINTS for getting correct information about the intersection point and 3 POINTS for getting correct information about B.

### Exam outcome

About 95 students took this exam. Several versions of this exam were given, with statistics for the versions fairly consistent (+/- 2 points). Overall, the mean grade achieved was 62.1, the median was 62, the standard deviation was 19.6, and the grades ranged from 23 to 98. A grade below 53 on this exam was unsatisfactory, equivalent to a letter grade of D or F, but in fact students with grades below 60 should be concerned about their progress in this course.