**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 (as in this
exam) exact values of certain functions are needed, such as in
problems 1, 2, 3 and 7. 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 method of
solution. Valid solutions of any type will be graded in a manner
similar to what is described below.

2 POINTS for differentiating the polynomial correctly. 3 POINTS for finding the critical numbers correctly. 4 POINTS for realizing that the extrema must occur at either the endponts or the critical number (of this, 2 POINTS for an explicit statement: "briefly explain" validated). 3 POINTS for giving the correct extreme values.

**2. (12 POINTS)**

a) 6 POINTS: 3 POINTS for the product rule and 2 POINTS for the chain
rule. 1 POINT for correct evaluation.

b) 6 POINTS: 4 POINTS for differentiating correctly (2 POINTS off if
the chain rule is not applied). Evaluating the derivative is 1 POINT,
and 1 POINT for the correct inference about the behavior of Q from the
sign of the derivative.

**3. (16 POINTS)**

2 POINTS for a picture (a picture need not be drawn -- if there is
successful work in the problem, these points will be given), 1 POINT
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, 2 POINTS for explicitly substituting
correctly and finding the two dimensions (and explaining which is
which!), and, finally, 3 POINTS for some explanation of why the answer
found provides a minimum area.

**4. (20 POINTS)**

a) 4 POINTS: 1 POINT for the domain. 1 POINT for where the function is
0. 2 POINTS each for the asymptotes.

b) 5 POINTS: 2 POINTS for the derivative computed correctly. 1 POINT
for detecting the critical number. 1 POINT for identifying the
critical number (as to rel max or min or neither) and 1 POINT for
explaining why.

c) 6 POINTS: 4 POINTS for computing and simplifying the second
derivative. 2 POINTS for correctly explaining why the statement is
correct (1 POINT for each assertion).

d) 5 POINTS: 3 POINTS for the picture and 2 POINTS for the labels
requested.

**5. (12 POINTS)**

a) 1 POINT for finding the surface area correctly.

b) 11 POINTS: 2 POINTS for the derivative computed and evaluated
correctly. 3 POINTS for the linear approximation formula applied
correctly. 1 POINT for using and converting percentages correctly
(there are two such applications in the problem). 3 POINTS for an
explanation of the discrepancy between the true answer and the
approximating answer, and 2 POINTS for either a correct supporting
second derivative computation or a correct supporting picture.

**6. (14 POINTS)**

7 POINTS for the graph (a correct picture must include the data about
the seven given points and the various limits). 1 POINT each for the
relative extreme and inflection points (a total of 4 POINTS). 1 POINT
each for the equations of the vertical and horizontal asymptotes (a
total of 3 POINTS).

**7. (14 POINTS)**

a) 7 POINTS: 5 POINTS for correct use of the chain and product
rules. 2 POINTS for solving for the derivative correctly.

b) 7 POINTS: 2 POINTS for "plugging in" to the original equation to
get one linear equation. 2 POINTS for "plugging in" to the derivative
(with the correct derivative information) to get another linear
equation. 3 POINTS for solving these equations correctly.