**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 problems 3 and 6. 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.

a) 8 POINTS: 2 POINTS each for the value (or in the case of the last, the non-existence) of the limit. Some evidence or discussion of the non-existence must be offered to earn the last 2 POINTS.

b) 4 POINTS: 2 POINTS for each point on the graph satisfying h´(x)=0: 1 POINT for declaring "Yes, here it is!" and 1 POINT for giving a credible value of h(x).

**2. (14 POINTS)**

5 POINTS: for the algebra involved in finding the first
(x) coordinates of the two intersection points.

2 POINTS: another point each given for finding the second (y)
coordinates of the two intersection points.

As written in the statement of the problem, valid algebraic techniques
(not trial-and-error or graphs drawn by a calculator) must be shown to
earn the 7 POINTS mentioned above.

7 POINTS given for the graph: 1 POINT given for each of the requested
labels: each of the points, the line, and the hyperbola. 1 POINT given
for the line (either an incorrect slope or y-intercept loses the
point). 2 POINTS given for the hyperbola, which should be in the
correct quadrants with correct asymptotic behavior.

**3. (16 POINTS)**

a) 5 POINTS: 4 POINTS for discussing how/why: some reasoning must be
given. 1 POINT for the correct value of A. We've studied continuity,
so the correct words and techniques (involving LIMITS) are available.

b) 4 POINTS: the graph should be continuous (!) otherwise -2 POINTS. 2
POINTS for the exponential curve, and 2 POINTS for the line segment. -2
points for graphing both functions over the entire interval.

c) 3 POINTS: 1 POINT for the correct answer, and 2 POINTS for some
reasoning or description of what's happening.

d) 4 POINTS: 2 POINTS for each domain and formula. The assignment of x=1
could be done in either "piece".

**4. (20 POINTS)**

Each part is worth 5 POINTS: the answer alone is worth 1 POINT, and
other work (how/why/explanation) is worth 4 POINTS. A graph can give
acceptable verification for parts c) and d); alternatively, some
analysis of asymptotic behavior must be given.

**5. (11 POINTS)**

a) 8 POINTS: 2 POINTS for the statement of the
definition of f'(x) (leaving out "lim" in the definition loses a
point!), and 6 POINTS for successfully manipulating the difference
quotient and getting the derivative. 0 POINTS for a correct answer
which is not supported by algebra.

b) 3 POINTS: 1 POINT for getting the slope of the line, 1 POINT for
getting the y-intercept or some point on the line, and 1 POINT for
giving a valid equation for the tangent line.

**6. (17 POINTS)**

a) and b) and c) 9 POINTS: The answers alone can be written with little
effort: 3 POINTS for each of these.

d) 8 POINTS: 2 POINTS for k(PI), and 3 POINTS for each of k´(PI) and
k´´(PI). The numerical values of each of these are worth 1 POINT, so
if a value is omitted, -1 POINT.

**7. (10 POINTS)**

2 POINTS for finding the slope of the line joining the 2 given
points. 2 POINTS for connecting this number with the derivative at
x=2. 1 POINT for the correct value of C. 2 POINTS for using the second
(y) coordinate of the given point in a correct equation, and 2
POINTS if the equation has the correct value for x in it. Finally 1
POINT for the correct value of D.