| Background about grading | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| Exam outcome | ||||||||||||||
| Show your work. Full credit may not be given for an answer alone. |
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.
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.
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."
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.
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.
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").
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).
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?"
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.
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.
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.
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.
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.
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").
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.
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).
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.