| Show your work. Full credit may not be given for an answer alone. |
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. Rarely in this exam values of certain functions are supposed to be "simplified", such as in problems 2 and 11b). 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.
2. (12 POINTS)
4 POINTS for each asymptote: 2 POINTS for each answer which should be
a correct equation (if the equation should be "x= 4 ln 7", then "4 ln
7" alone loses 1 POINT), and 2 POINTS for showing SOME WORK. In this
problem, computer evidence, since it is not specifically ruled out, is
acceptable. But note again on the cover page: "An answer alone may not
get full credit." -1 POINT for not correctly simplifying the exp/ln
computation.
3. (12 POINTS)
2 POINTS for the statement of the definition of f´(x) (leaving
out "lim" in the definition loses 1 POINT). 10 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 10 POINTS:
3 POINTS for validly replacing the definition by the specific function given,
and 4 POINTS for doing valid algebra on it, and 3 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.
4. (14 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) 4 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. You can only lose 4 POINTS, though, in this
section!
c) 4 POINTS: 2 POINTS for the correct answer (that the function is NOT
differentiable at the two suspect points) and 2 POINTS for some
explanation (e.g., the graph has a corner, or the difference quotient
limit doesn't exist).
5. (12 POINTS)
Each part is worth 4 POINTS. In each part, -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.
6. (10 POINTS)
a) 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.
b) 4 POINTS for some valid equation of the tangent line. 2 POINTS for
getting the slope correct (substituting x and y values in to get
y´). -1 POINT for leaving a quotient in the algebraic equation for
a line instead of multiplying out.
7. (20 POINTS)
a) 2 POINTS: 1 POINT for a valid equation connecting b and h and 1 POINT for
getting the length of h.
b) 7 POINTS: 4 POINTS for differentiating the equation correctly, and 2
POINTS for solving for the derivative with the correct numerical answer,
which need NOT be simplified. 1 POINT for correctly responding to the
{in/de}creasing question.
c) 2 POINTS: 1 POINT for a valid equation connecting b, h, and theta,
and 1 POINT for getting theta. A correct answer could be arctan (4/3)
or it could be a numerical approximation using radians OR degrees.
d) 9 POINTS: 6 POINTS for differentiating the equation linking the
sides and the angle correctly, including getting theta´. -2 POINTS
each for omitting theta´ or for applying the chain rule
incorrectly on either side. 2 POINTS for substituting in and getting
the correct numerical value of theta´, and 1 POINT for correctly
responding to the {in/de}creasing question.
8. (26 POINTS)
a) 5 POINTS: These instructions are for version A where the derivative
has two maximal open intervals indicated where it is positive and one
where it is negative. Other versions will be graded in a similar
fashion. 2 POINTS for saying where the function is increasing, and 1
POINT for saying why (sign of h´): we accept two adjoining
intervals or one whole interval, and will also accept for reasoning
h´ positive for the whole interval even though non-negative is
sufficient. 1 POINT for saying where the function is decreasing, and 1
POINT for saying why (sign of h´).
b) 5 POINTS: 2 POINTS for the up-down-up nature of the graph and 3 POINTS for
having 3 roots in (approximately) the correct places.
c) 6 POINTS: 2 POINTS for saying where the function is concave up and 1 POINT
for saying why (sign of h´´). 2 POINTS for saying where the function is
concave down and 1 POINT for saying why (sign of h´´).
d) 4 POINTS: 1 POINT for the relative extrema and its location, and 3
POINTS for the inflection points and their location.
e) 6 POINTS: 4 POINTS for each requested labeled point (there are
three inflection points and one relative extremum) and 2 POINTS for
the general shape of the curve. Please note that the points MUST be
labeled ON the graph, and if they are not, the credit is NOT earned.
Reason: credit was already earned for detecting the values of x in the
section above. This section's answer tries to detect if students know
what the picture of each behavior looks like.
9. (14 POINTS)
2 POINTS for the constraint, 1 POINT for the constraint solved for one
variable, 2 POINTS for the objective function, 1 POINT for reducing
the objective function to a function of one variable, 2 POINTS for
differentiating the objective function correctly, 2 POINTS for finding
out where the derivative is 0, 1 POINT for explicitly substituting
correctly and finding L and W, and, finally, 3 POINTS for some
explanation of why the answer found provides a minimum (note that ANY
explanation using function limits or first derivative behavior or
second derivative value is fine).
10. (10 POINTS)
a) 5 POINTS: 1 POINT for F(the number) and 4 POINTS for F´(the
number): 2 POINTS for a correct algebraic equation and 2 POINTS for
evaluating the expressions correctly.
b) 5 POINTS: 3 POINTS for the correct linear approximation equation
(OR indicating that the student knows it by using it), and 2 POINTS
for filling in the numbers correctly. NO arithmetic need be done, but
any arithmetic done must be done correctly, else -1 POINT (see top
instructions).
11. (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 simplifying the exp/ln computation each time.
12. (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.
13. (14 POINTS)
4 POINTS for the sketch: parabola opening down and symmetry, with
approximately correct x and y intercepts. 4 POINTS for writing the
correct definite integral and 6 POINTS for correct use of the
Fundamental Theorem of Calculus.
14. (16 POINTS)
a) 8 POINTS: 5 POINTS for the correct antiderivative and then
3 POINTS for the correct answer (which doesn't need to be "simplified").
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.
15. (12 POINTS)
a) 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, but numerical
values using an approximation to pi should be accepted.
b) 4 POINTS for the answer alone but just 2 POINTS if the function value is
NOT evaluated: e.g., H(the correct number).
c) 4 POINTS. The answer need not be computed entirely, but values of H
should be inserted, otherwise -2 POINTS (the implication here is that
H and Q can't be distinguished).
Discussion of the grades
A very large number 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 193, with a median grade of 100 and a mean grade of 100.8. The standard deviation was 44.7. 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 | 64 | F | 22.9 |
| 65 | 84 | D | 14.6 |
| 85 | 109 | C | 20.7 |
| 110 | 124 | C+ | 10.3 |
| 125 | 139 | B | 9.9 |
| 140 | 154 | B+ | 7.2 |
| 155 | 200 | A | 14.7 |
The exam was difficult but the instructional staff feels that this difficulty is acknowledged in the suggested letter grades. Realize also that these grades include the work of all students registered for the course who show up at the final. Math 135 instructors can verify that many students who took the final exam have not participated in the course in any meaningful way for much of the semester. Such students' grades are included in the data reported above.
There were several versions of the final exam. Grades on the versions seem to be statistically similar.