## Grading guidelines for the second exam

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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. The statements of the questions should be a guide to whether an approximation is requested or allowed.

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 (8 POINTS)
1 POINT for y. 4 POINTS for the chain rule applied correctly. 3 POINTS for the correct value of dy/dt. -1 POINT for not giving the exact value of one or both trigonometric functions (the point is just taken off once!).

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Problem 2 (10 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 part b); alternatively, some comment on the behavior of the function must be given.

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Problem 3 (12 POINTS)
3 POINTS for correct differentiation, 4 POINTS for finding the critical numbers, 2 POINTS for saying what each one is, and 3 POINTS for some explanation.

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Problem 4 (14 POINTS)
a) 1 POINT.
b) 7 POINTS: 5 POINTS for correct use of the chain and product rules. 2 POINTS for solving for the derivative correctly.
c) 5 POINTS: 2 POINTS 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!
d) 1 POINT.

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5. (16 POINTS)
2 POINTS for the objective function (the area), and 2 POINTS for writing it as a function of one variable using the equation. 4 POINTS for differentiating the objective function correctly, 3 POINTS for finding out where the derivative is 0, and 2 POINTS for explicitly stating with identification what the sides of the rectangle with largest area are. Finally, 3 POINTS for some explanation of why the answer found provides a maximum (note that ANY explanation using function values at endpoints or first derivative behavior or second derivative value is fine but SOME EXPLANATION must be explicitly given).

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6. (20 POINTS)
a) 9 POINTS: 2 POINTS for computing W´(x) and 2 POINTS for computing W´´(x). 3 POINTS for solving W´(x)=0 and 2 POINTS for solving W´´(x)=0.
b) 1 POINT.
c) 10 POINTS: 2 POINTS for the graph and 3 POINTS for the labels and 5 POINTS for the answers. Note that the domain of W(x) is x>0, and reference to intervals not in the domain will be penalized by loss of 1 POINT.

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7. (20 POINTS)
a) 8 POINTS: 0 POINTS for confusing the function and the derivative. 1 POINT for locating the correct x (+/- .25 accuracy is good enough). 3 POINTS for indicating it is a relative extremum. 4 POINTS for giving enough explanation to conclude that the desired extreme value of the function actually occurs at the number specified.
b) 12 POINTS: 3 POINTS for the linear approximation formula cited and applied correctly. 2 POINTS for the derivative computed and evaluated correctly. 1 POINT for the actual numerical answer. 3 POINTS for an explanation of the discrepancy between the true answer and the approximating answer involving the second derivative (these points can also be earned for a picture of the function near the approximation together with its tangent line). 3 POINTS for getting the necessary information about the second derivative and its sign.

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### Exam outcome

More than 90 students took this exam. Several versions of this exam were given, with statistics for the versions very close. Overall, the mean grade achieved was 46.14, the median was 46, the standard deviation was 20.05, and the grades ranged from 8 to 94. A grade below 45 on this exam was unsatisfactory, equivalent to a letter grade of D or F.

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