## Grading guidelines for the first exam

1 2 3 4 5 6 7
Exam outcome

### 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 the students' 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, 4 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.

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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.

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Problem 2 (14 POINTS)
a) 2 POINTS for the statement of the definition of f´(x) (leaving out "lim" in the definition loses 1 POINT!).
b) 8 POINTS for successfully manipulating the difference quotient and getting the derivative. 2 POINTS of these are for inserting x+delta x successfully in the formula for f. 0 POINTS for a correct answer which is not supported by algebra.
c) 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!

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Problem 3 (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 parabolic curve, and 2 POINTS for each correctly drawn line segment. If the parabolic curve bends perceptibly, it should bend in the correct way, else -1 POINT. -2 POINTS for graphing several functions over the same interval.

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Problem 4 (16 points)
a) 6 POINTS: a bald answer (with no justification) is acceptable here. 2 POINTS for asserting that the length of BC is 5-x, and 2 POINTS more each for finding the area of each square.
b) 4 POINTS: the graph should be the graph of the function in a) and should fit entirely within the "window" provided, and should never be 0. Such a graph should get 2 POINTS. If the graph goes markedly outside of the window, -1 POINT. For the second 2 POINTS, the graph should look like a parabola opening up, its minimum should be almost halfway up, and it should be reasonably symmetric vertically (around the line x=2.5).
c) 6 POINTS: 4 POINTS for showing some algebraic work (2 POINTS of these for successfully "expanding" (x-5)2) and 1 POINT each for finding valid x's.

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Problem 5 (9 POINTS)
Each part is worth 3 POINTS. The answers alone can be written with little effort. -1 POINT for a simple error (miswritten number, for example). -2 POINTS for a product/chain/quotient rule error.

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Problem 6 (8 POINTS)
2 POINTS for S(0), and 3 POINTS for each of S´(0) and S´´(0). The numerical values of each of these are worth 1 POINT, so -1 POINT for any omitted value.

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Problem 7 (19 POINTS)
a) 5 POINTS: 1 POINT each. There should be five numbers given.
b) 4 POINTS: 1 POINT for each interval, consistent with the numbers in part a). -1 POINT if an endpoint (NOT 2!) is included in one of the intervals, and an additional -1 POINT if 2 is included in one of the intervals.
c) 1 POINT for the answer.
d) 1 POINT for the answer.
e) 1 POINT for the answer.
f) 2 POINTS: 1 POINT each. There should be two numbers given.
g) 2 POINTS: 1 POINT each. There should be two numbers given, and the answers should be consistent with the numbers in part f).
h) 3 POINTS: 1 POINT for each interval, consistent with the numbers in part f).
POINTS can be taken off for more answers than are correct. For example, the answer "All numbers" in h) will earn 0 POINTS.

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### Exam outcome

About 100 students took this exam. Several versions of this exam were given, with statistics for the versions very consistent. Overall, the mean grade achieved was 56.4, the median was 55, the standard deviation was 18.43, and the grades ranged from 15 to 100. A grade below 50 on this exam was unsatisfactory, equivalent to a letter grade of D or F.