**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. Numerical answers
gotten from graphical information should be reasonably close. Thus in
problem 7, the third critical number should be about 1.9, and 2 or 22
would not be reasonably close.

**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.

5 POINTS for differentiating correctly, and then 2 POINTS for solving for the derivative correctly. 1 POINT off for minor errors, and 2 POINTS off for misuse of the chain or product rules. 2 POINTS for writing some correct equation for the tangent line.

Some students seem to have solved for y explicitly. So for this solution, the following point allocation is given: 2 POINTS for solving explicitly and correctly for y as a function of x. 5 POINTS for correct differentiation and correct evaluation. 2 POINTS for writing some correct equation for the tangent line.

**2. 10 POINTS**

2 POINTS for differentiating the polynomial correctly. 2 POINTS for
finding the critical numbers. 4 POINTS for realizing the the max/min
must occur either at the endpoints or the critical number. 2 POINTS
for giving the correct extreme values.

**3. 21 POINTS**

a) 1 POINT for the correct root.

b) 3 POINTS: 2 POINTS for the derivative computed correctly and 1
POINT for the correct root of the derivative.

c) 3 POINTS: 2 POINTS for the second derivative computed correctly and
1 POINT for the correct root of the second derivative.

d) 14 POINTS: 6 POINTS for the algebraic answers distributed as
follows: 1 POINT for the interval of increase; 1 POINT for the
interval of decrease; 1 POINT for the interval which is concave up and
1 POINT for the interval which is concave down; 1 POINT for the
correct pair of numbers; 1 POINT for the correct pair of numbers.
8 POINTS for the sketch: 1 POINT for each label (6 POINTS total) and 2
POINTS for the correct picture.

**4. 10 POINTS**

a) 3 POINTS for giving the correct formula.

b) 7 POINTS: 1 POINT for finding theta at the specified time. 3 POINTS
for correctly differentiating the formula in a) with respect to
time. 1 POINT for correctly finding (sec theta)^{2} squared at
the specified time, 1 POINT for evaluating the right-hand side
correctly, and 1 POINT for getting the final answer correctly.

**5. 14 POINTS**

4 POINTS for writing the correct area formula. 3 POINTS for
differentiating it correctly. 2 POINTS for finding the critical
number. 2 POINTS for finding the dimensions of the rectangle. 3 POINTS
for saying why the rectangle is found has maximum area.

**6. 12 POINTS**

a) 3 POINTS for finding the derivative correctly. 3 POINTS for writing
and using the approximation formula correctly.

b) 3 POINTS for an explanation, and 3 POINTS for a correct supporting
computation.

**7. 24 POINTS**

a) 3 POINTS: 1 POINT for each critical number.

b) 6 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) 4 POINTS: 1 POINT for each interval correctly named.

d) 3 POINTS: 1 POINT for each inflection point.

e) 8 POINTS: 5 POINTS for labels (1 POINT each), and 3 POINTS for the
curve.