## Grading guidelines for the Math 135 final exam in spring 1999

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Exam outcome

### Background

The cover page states:
 Show your work. Full credit may not be given for an answer alone.
Therefore a "bald" answer with no supporting work may get little if any credit.

Arithmetic errors will be penalized in the following way: -1 for the first error, and -1 for any additional arithmetic errors in that problem. 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 in this exam values of certain functions (such as exp, ln, and trig functions) are supposed to be "simplified" by evaluation. This occurs in problems 2, 11, 12c), and 14. In each case, the problem statements specifically request such answers.

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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1. (10 POINTS)
a) 4 POINTS for algebraic analysis (e.g., dividing top and bottom by x2, or writing that the dominant terms top and bottom are the x2 terms), and 1 POINT for the answer.
b) 4 POINTS for correct algebraic manipulation, and 1 POINT for the answer.

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2. (12 POINTS)
4 POINTS for each asymptote and work on it.
Credit for each asymptote is assigned as follows: 2 POINTS for each answer which should be a correct equation (if the equation should be "x= 17", then "17" alone loses 1 POINT), and 2 POINTS for showing SOME WORK. In this problem, calculator evidence, since it is not specifically ruled out, is acceptable. But note again the cover page's declaration: "An answer alone may not get full credit."

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3. (18 POINTS)
a) 10 POINTS: 2 POINTS for the statement of the definition of f´(x) (leaving out "lim" in the definition loses 1 POINT). 8 POINTS for successfully manipulating the difference quotient and getting the derivative, but 0 POINTS for a correct answer which is not supported by algebra. As to the other 8 POINTS: 3 POINTS for validly replacing the definition by the specific function given, and 3 POINTS for doing valid algebra on it, and 2 POINTS for taking the limit correctly. POINTS to be taken off as described in the general comments for arithmetic errors, with more taken off for algebraic errors.
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!
c) 4 POINTS: 2 POINTS for a correct piece of parabolic arc. -1 POINT for the arc curved the wrong way. 2 POINTS for a tangent line segment. 0 POINTS if the line segment is not close to tangent.

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4. (14 POINTS)
Parts a) and b) are each worth 4 POINTS. In each of these parts, -1 POINT for a minor error, and -2 POINTS for misuse of the chain/product/quotient rules. The last includes incorrect cancellations or combinations in and out of functions.
c) 6 POINTS (2 POINTS for successfully solving the differentiated equation for the derivative, -2 POINTS each for misuse of the product rule or chain rule). It is still possible to get 2 POINTS by solving for y´, though, even with a "broken" equation if y´ appears non-trivially, that is, at least twice in the equation with non-constant coefficients.

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5. (16 POINTS)
a) 6 POINTS: 1 POINT for giving the answer, and 5 POINTS for some explanation or computation.
b) 6 POINTS: the graph should be continuous, otherwise -2 POINTS. 2 POINTS for the general shape (how it {in/de}creases: 1 POINT off for each error in this behavior). If not the graph of a function (that is, it is not "single-valued"), -2 POINTS.
c) 4 POINTS: the answer alone is worth 1 POINT. 3 POINTS for some explanation ("The graph has a corner" or "The left- and right-hand limits defining the derivative don't coincide").

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6. (18 POINTS)
a) 6 POINTS: 3 POINTS for analysis of N´ and 3 POINTS for analysis of N´´.
b) 6 POINTS: 1 POINT for locating the relative maximum and 1 point for its value and 1 point for some justification. 1 POINT for locating the relative minimum and 1 point for its value and 1 point for some justification. No additional points for "rejecting" the inflection point, but -2 POINTS if it is included as a relative extremum. The justification could gotten using calculator evidence, since such was not specifically excluded.
c) 6 POINTS: 1 POINT for location of each inflection point and 1 POINT for some justification. Again, calculator evidence could be cited (a correct window is not too hard to find).

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7. (12 POINTS) 3 POINTS for writing a correct formula for A, the area of the annular region. 1 POINT for finding A "at that time". 4 POINTS for differentiating the formula correctly. 3 POINTS for getting the numerical answer to "How fast is A changing at that time". 1 POINT for answering "Is A increasing or decreasing at that time?"

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8. (16 POINTS)
5 POINTS for the objective function (2 of these can be given for drawing an appropriate figure), 3 POINTS for differentiation of the function, 3 POINTS for finding the critical numbers, 2 POINTS for finding the ``largest possible volume'', and 3 POINTS for some explanation (e.g., endpoint analysis) of why the answer does give a maximum.

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9. (18 POINTS)
5 POINTS (1 POINT each) for correct labeling on the graph of m and M and I. 9 POINTS (1 POINT each) for correcting giving the intervals requested. 4 POINTS for the graph of h(x): the curve should be the graph of a continuous function (2 POINTS) whose domain is all x except 2 (1 POINT), and it should have appropriate asymptotic behavior at 2 (1 POINT).
If a wrong graph is drawn, then all or most of the graph points may be lost. If it is correctly labeled and then correct conclusions are drawn about increase and decrease and concavity, the grader may give additional points.

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10. (8 POINTS)
a) 4 POINTS for a correct derivative.
b) 4 POINTS for a clear and correct explanation, which may be brief. For example, the answer "Q´ is always positive between A and B, so Q(A) < Q(B)" (with the relevant A and B) would get full credit. Or there could be some assertion about critical numbers and the sign of the derivative at one point in the interval, etc.

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11. (8 POINTS)
2 POINTS for the first correct antiderivative and 2 POINTS for evaluating the first constant of integration correctly. 2 POINTS for the second correct antiderivative and 2 POINTS for evaluating the second constant of integration correctly. -1 POINT for not evaluating the ln function correctly.

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12. (12 POINTS)
a) 3 POINTS for getting the correct derivative.
b) 4 POINTS for getting the correct values of P and Q: some explanation must be given or else -2 POINTS.
c) 5 POINTS: 3 POINTS for using the Fundamental Theorem correctly: basically realizing the connection between the result of b) and this. 2 POINTS for evaluating the antiderivative correctly and completely.

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13. (12 POINTS)
a) 4 POINTS for the sketch. The parabola should open down and intersect the x-axis correctly.
b) 8 POINTS: 2 POINTS for using the intersection with the x-axis correctly to set up the definite integral, 2 POINTS for writing the area as a definite integral, and 4 POINTS for completing the problem by integrating and substituting (the answer need not be "simplified").

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14. (16 POINTS)
a) 8 POINTS: 5 POINTS for the correct antiderivative. 3 POINTS for the correct answer.
b) 8 POINTS: 5 POINTS for the correct antiderivative but if something blatantly horrid is written (e.g., integration as a multiplicative operation), 0 POINTS. -2 POINTS for an error of a multiplicative constant in the antiderivative. 3 POINTS for the correct answer.

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15. (10 POINTS)
a) 4 POINTS. The answer need not be computed entirely, but values of R should be inserted, otherwise -2 POINTS.
b) 4 POINTS for the answer: 2 POINTS if there is a difference of areas (as there should be), and 2 POINTS for writing the correct answer numerically. The answer doesn't need to be simplified.
c) 2 POINTS for the answer alone but just 1 POINT if the function value is NOT evaluated: e.g., R(the correct number).

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Approximately 700 of the final exams were graded in common using the guidelines above. The following information applies to those exams.

The grades ranged from 4 to 200, with a median grade of 103 and a mean grade of 103.07. The standard deviation was 35.8. A recommended translation from numerical final exam grade to letter final exam grade, along with the percentage of the students who got these grades are as follows:

 Numerical score Letter grade equivalent Percentage of all grades Low # High # 0 89 F 35.8 90 99 D 10.7 100 114 C 16 115 129 C+ 10.6 130 144 B 11 145 159 B+ 7.6 160 200 A 8

There were several versions of the final exam. The statistics for the two versions seem to be quite similar.

## What happens to the papers?

Rutgers regulations require instructional units to keep final exams for one year. Students who wish to inspect their exams should make appropriate arrangements with their instructors or with the Undergraduate Office of the Math Department.

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