**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) values of certain functions are supposed to be "simplified",
such as in problem 2. The statements of the questions should be a
guide to that. In other questions, algebraic manipulation of
expressions is almost essential to success (for example, problem 4c).

**Other methods** than are given in the "official" answers may be
good strategies for these problems so the answers presented are not
the only valid solutions. Any correct solutions will be graded in a
manner similar to what is described below.

In each case, 1 POINT off for minor errors, and 2 POINTS off for misuse of the chain/product/quotient rules. The last includes incorrect cancellations or combinations in and out of functions.

a) 5 POINTS

b) 5 POINTS

c) 7 POINTS (2 POINTS for successfully solving the differentiated equation for the derivative).

**2. (17 POINTS)**

a) 2 POINTS: just the answers.

b) 2 POINTS: again, just the answers, but as required by the statement
of the problem.

c) 4 POINTS: 2 POINTS for the correct derivative, and 2 POINTS for
detecting the root of G'(x) (1 POINT reserved for correct statement of
the answer).

d) 4 POINTS: 2 POINTS for the correct second derivative (based on the
student's first derivative) , and 2 POINTS for detecting the root of
G''(x) (1 POINT reserved for correct statement of the answer).

e) 5 POINTS: 2 POINTS for the picture, and 3 POINTS for each labelled
feature (the inflection point, the relative minimum, and the
horizontal asymptote).

**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 S and H, and, finally, 3 POINTS for some
explanation of why the answer found provides a minimum area. I think
this is a problem from the textbook.

**4. (22 POINTS)**

a) 4 POINTS: 2 POINTS for the roots of F', and 1 POINT
each for identifying where F'>0 and F'<0.

b) 4 POINTS: 1 POINT each for identifying each critical number (as to
rel max or min or neither) and 1 POINT each for telling why. 0 POINTS
for just giving the critical numbers which would duplicate what was
earned in a).

c) 5 POINTS: 2 POINTS for a correct expression for F'' and 1 POINT
each for roots of F''.

d) 3 POINTS: 1 POINT each for indentifying each of the two points of
inflection, and 1 POINT for telling why they are points of inflection.

e) 6 POINTS: 3 POINTS for the picture, and 3 POINTS for identifying
the two points of inflection and the relative minimum. Drawing this
graph to show the essential features is not easy.

**5. (15 POINTS)**

7 POINTS for getting an area formula (2 POINTS of these
for declaring a correct area formula for a triangle). 3 POINTS for
finding the first derivative and the critical number for the area
formula. 2 POINTS for giving the vertices of the triangle. 3 points
for saying why the triangle found has maximum area.

**6. (13 POINTS)**

a) 3 POINTS for giving the correct formula.

b) 10 POINTS: 1 POINT for finding theta at the specified time. 4
POINTS for correctly differentiating the formula in a) with respect to
time. 1 POINT for finding (sec theta) squared at the specified time,
and 2 POINTS for computing the right hand side at that time. 1 POINT
for solving for the requested derivative correctly, and 1 POINT for
declaring that it is decreasing.