## 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, show 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 interested in seeing how students solve non-formulaic problems.

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 6 and 7. The statements of the questions should be a guide to that.

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.

Partial credit was assigned according to the recipes below. Some of the instructors in this course may allocate partial credit more strictly than what's indicated here.

### Discussion of grading for each problem

1. (16 POINTS)
6 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.
8 POINTS given for the graph: 1 POINT given for each of the requested labels: each of the points, the line, and the parabola. 2 POINTS given for the line, 1 each for slope and for y-intercept. 2 POINTS given for the parabola, which should open "up", have a vertex at (0,-1), and have approximately correct x-intercepts.

2. (16 POINTS)
a) 3 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 "cubic" curve, and 2 POINTS for the line segment. -2 points for graphing both functions over the entire interval.
c) 4 POINTS: 2 POINTS for each domain & formula. The assignment of x=1 could be done in either "piece".
d) 4 POINTS: this should be a reflection across y=x of the answer to b), otherwise -2 POINTS. As in b), the graph should be continuous, with the same assignment of point values for each piece.

3. (20 POINTS)
I think these questions were all copied from assigned homework problems! 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 part d).

4. (10 POINTS)
The first part of the problem is worth 4 POINTS. Deductions: 1 POINT off for inverting the fraction, 2 POINTS off for writing tan instead of arctan, and 2 POINTS off for incorrect cancellations in and out of the functions. The second part of the problem is worth 1 POINT. The last part of the problem is worth 5 POINTS. Again, 1 POINT off for inverting a fraction incorrectly and 2 POINTS off for incorrect cancellations in and out of the functions. 2 POINTS for writing a correct equation for this problem involving sin.

5. (16 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) 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.
c) 5 POINTS: 1 POINT each for the requested labels. 1 POINT for the straight line, 1 POINT for the curve, and 1 POINT for their tangency (they should touch "tangently"!) at the correct point of the graph.

6. (12 POINTS)
Certainly for a) and c) the answers alone can be written with little effort: 3 POINTS for each of these. Part b) is worth 4 POINTS, and part d) is worth 2 POINTS, one for each value.

7. (10 POINTS)
a) 2 POINTS: 1 POINT for drawing the tangent line, and 1 POINT for the numerical answer.
b) 8 POINTS: 2 POINTS for f'(x). 2 POINTS for solving where f'(x)=0 and getting the correct answer, 1 POINT for plugging in f(ln 3), 1 POINT for expressing the answer suitably, and, finally, 2 POINTS for stating what the coordinates of A are.

### Exam outcome

About 100 students took this exam. Several versions of this exam were given, with statistics for the versions remarkably consistent. Overall, the mean grade achieved was 67.8, the median was 69, the standard deviation was 18.7, and the grades ranged from 24 to 99.