## 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) values of certain functions are supposed to be "simplified", such as in problems 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. (12 POINTS)
a) 8 POINTS: 2 POINTS each for the value (or in the case of the last, the non-existence) of the limit. Some evidence or discussion of the non-existence must be offered to earn the last 2 POINTS.
b) 4 POINTS: 2 POINTS for each point on the graph satisfying h´(x)=0: 1 POINT for declaring "Yes, here it is!" and 1 POINT for giving a credible value of h(x).

2. (14 POINTS)
5 POINTS: for the algebra involved in finding the first (x) coordinates of the two intersection points.
2 POINTS: another point each given for finding the second (y) coordinates of the two intersection points.
As written in the statement of the problem, valid algebraic techniques (not trial-and-error or graphs drawn by a calculator) must be shown to earn the 7 POINTS mentioned above.
7 POINTS given for the graph: 1 POINT given for each of the requested labels: each of the points, the line, and the hyperbola. 1 POINT given for the line (either an incorrect slope or y-intercept loses the point). 2 POINTS given for the hyperbola, which should be in the correct quadrants with correct asymptotic behavior.

3. (16 POINTS)
a) 5 POINTS: 4 POINTS for discussing how/why: some reasoning must be given. 1 POINT for the correct value of A. We've studied continuity, so the correct words and techniques (involving LIMITS) are available.
b) 4 POINTS: the graph should be continuous (!) otherwise -2 POINTS. 2 POINTS for the exponential curve, and 2 POINTS for the line segment. -2 points for graphing both functions over the entire interval.
c) 3 POINTS: 1 POINT for the correct answer, and 2 POINTS for some reasoning or description of what's happening.
d) 4 POINTS: 2 POINTS for each domain and formula. The assignment of x=1 could be done in either "piece".

4. (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 analysis of asymptotic behavior must be given.

5. (11 POINTS)
a) 8 POINTS: 2 POINTS for the statement of the definition of f'(x) (leaving out "lim" in the definition loses a 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) 3 POINTS: 1 POINT for getting the slope of the line, 1 POINT for getting the y-intercept or some point on the line, and 1 POINT for giving a valid equation for the tangent line.

6. (17 POINTS)
a) and b) and c) 9 POINTS: The answers alone can be written with little effort: 3 POINTS for each of these.
d) 8 POINTS: 2 POINTS for k(PI), and 3 POINTS for each of k´(PI) and k´´(PI). The numerical values of each of these are worth 1 POINT, so if a value is omitted, -1 POINT.

7. (10 POINTS)
2 POINTS for finding the slope of the line joining the 2 given points. 2 POINTS for connecting this number with the derivative at x=2. 1 POINT for the correct value of C. 2 POINTS for using the second (y) coordinate of the given point in a correct equation, and 2 POINTS if the equation has the correct value for x in it. Finally 1 POINT for the correct value of D.

### Exam outcome

About 100 students took this exam. Several versions of this exam were given, with statistics for the versions quite consistent. Overall, the mean grade achieved was 52.6, the median was 52, the standard deviation was 18.0, and the grades ranged from 14 to 92. Numerical grades below 47 translated into unsatisfactory letter grades (D's and F's).