### 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 situation which is more difficult 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". This occurs in problems 9b), 12b), 13 and 15a). 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.

### Discussion of grading for each problem

1. (15 POINTS)
2 POINTS for the objective function (the area), and 2 POINTS for writing it as a function of one variable using the equation. 3 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).

2. (12 POINTS)
a) 6 POINTS: 3 POINTS for writing correct conditions connecting A and B with numbers (take 2 POINTS off for the first mistake in an equation), and 3 POINTS for solving the system of equations written by the student correctly.
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.

3. (8 POINTS)
a) 3 POINTS for correct algebraic manipulation, and 1 POINT for the answer.
b) 3 POINTS for algebraic analysis (e.g., dividing top and bottom by x3, or writing that the dominant terms top and bottom are the x3 terms), and 1 POINT for the answer.

4. (12 POINTS)
4 POINTS for each asymptote.
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."

5. (16 POINTS)
2 POINTS for writing a correct equation connecting X and Y. 3 POINTS for correctly differentiating this equation or a related one (after all, the equation can be "solved" for Y as a function of X with little difficulty). 1 POINT for finding the correct value of Y for the given value of X. 2 POINTS for finding the correct value of y´ "at this time". The answer need not be simplified. 2 POINTS for writing a correct equation relating theta to some of the variables X and Y. 3 POINTS for correctly differentiating it. 1 POINT for finding the correct value of theta. The answer can be in degrees or radians. 2 POINTS for finding the correct value of theta´ "at this time". The answer need not be simplified.

6. (12 POINTS) The curve sketched should be the graph of a function whose domain is all x except c (1 POINT). The curve should have asymptote +infinity as x approaches c- and asymptote -infinity as x approaches c+ (2 POINTS). The curve should cross the x-axis in the appropriate 3 places (3 POINTS). It should have the correct asymptotic behavior as x approaches -infinity and as x approaches +infinity (2 POINTS). Finally, the behavior in the intervals between critical numbers and asymptotic values should be appropriate (4 POINTS), which includes indication of max/min of f´ (the inflection points of f).

7. (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.

8. (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.

9. (13 POINTS)
a) 6 POINTS: 3 POINTS for computing the function value, and 3 POINTS for computing the value of the derivative. Arithmetic errors lose a point each, as in the general directions.
b) 7 POINTS: writing a correct and appropriate version of the chain rule is 5 POINTS, and getting the correct numerical answer is 2 POINTS.

10. (14 POINTS)
The graph drawn should be a graph of a continuous, smooth function (2 POINTS). The inflection points and extrema should be correctly indicated (5 POINTS). The {in/de}creasing behavior should be correct (3 POINTS). The concavity should be correct (4 POINTS).

11. (8 POINTS)
The answers are worth 1 POINT each and need not be simplified. The justification is worth 6 POINTS: a computation of f´, some discussion of its sign, and the relevance of this to the conclusion must be given. Alternatively, the student could assert explicitly that there is no critical number within the interval, and that therefore extreme values must occur at the endpoints.

12. (12 POINTS)
a) 4 POINTS for getting the correct derivative.
b) 8 POINTS: 4 POINTS for using the Fundamental Theorem correctly: basically realizing the connection between the result of a) and this. 4 POINTS for evaluating correctly and completely. -1 POINT for not correctly evaluating the exponential function each time.

13. (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 (up to -2 POINTS) for not evaluating the trig functions correctly each time.

14. (20 POINTS)
a) 4 POINTS for the sketch. The parabola should open down and intersect the x-axis correctly. The horizontal line should intersect the parabola correctly.
b) 16 POINTS: Undoubtedly there will be many approaches to this part of the problem. One way to do the problem is to realize it is the difference between the area of a region with a curved boundary with sides the x-axis and two vertical lines and a rectangular region. 12 POINTS for finding the area of the region with the curved boundary, and 4 points for subtracting off the correct rectangular area.

15. (16 POINTS)
a) 8 POINTS: 5 POINTS for the correct antiderivative and then 3 POINTS for the correct answer.
b) 8 POINTS: -1 POINT if no "+ C" with an otherwise correct answer. If something blatantly horrid is written (e.g., integration as a multiplicative operation), 0 POINTS. -2 POINTS for each error of a multiplicative constant.

16. (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).

A large number (much more than one thousand) of the final exams were graded in common using the guidelines above. The following information applies to those exams.

The grades ranged from 0 to 198, with a median grade of 103 and a mean grade of 104.54. The standard deviation was 45.34. A recommended translation from numerical final exam grade to letter final exam grade along with the percentage of the students who got these grades is as follows:

 Numerical score Letter grade equivalent Percentage of all grades Low # High # 0 74 F 29.7 75 89 D 10.5 90 109 C 14.4 110 129 C+ 12.2 130 144 B 9.9 145 159 B+ 9.4 160 200 A 13.9

There were several versions of the final exam.

## 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.