**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 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.

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 comment on the behavior of the function must be given.

**2. (14 POINTS)**

a) 8 POINTS: 6 POINTS for discussing how/why: some reasoning must be
given. 1 POINT each for the correct values of A and B. We've studied
continuity, so the correct words and techniques (involving LIMITS) are
available.

b) 6 POINTS: the graph should be continuous (!) otherwise -2 POINTS. 2
POINTS for the correctly drawn cosine curve (please note that -Pi/2 is
between -2 and -1), and 2 POINTS for each correctly drawn line
segment. -2 points for graphing several functions over the same
interval.

**3. (12 POINTS)**

Each part is worth 6 POINTS: 4 POINTS for the formula, and 2 POINTS
for the domain. The correct endpoints for the domain must be given,
but either endpoint may be included/excluded from the domain because
the physical setting of "A ladder ..." can really be seen in various
ways.

**4. (12 POINTS)**

a) 8 POINTS: 2 POINTS for the statement of the definition of
f´(x) (leaving out "lim" in the definition loses 1 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) 4 POINTS: 1 POINT for getting the slope of the line, 1 POINT for
getting the y-intercept or some point on the line, and 2 POINTS for
giving a valid equation for the tangent line. -1 POINT for presenting
an equation (such as (y-y_{0}) DIVIDED by (x-x_{0}) =
m) which is not satisfied by EVERY point on the line!

**5. (9 POINTS)**

Each part is worth 3 POINTS. The answers alone can be written with little
effort.

**6. (8 POINTS)**

2 POINTS for j(0), and 3 POINTS for each of j´(0) and
j´´(0). The numerical values of each of these are worth 1
POINT, so if a value is omitted, -1 POINT.

**7. (13 POINTS)**

a) 2 POINTS: there are three places where Q is 0. -1 POINT each for
missing any one of them (up to -2 POINTS, of course).

b) 4 POINTS: there are two intervals where Q is positive. 1 POINT for
each correct end point of an interval, correctly presented. If an end
point is included in an otherwise correct answer, -1 POINT (taken off
only once in this section of the problem).

c) 3 POINTS: there are four places where Q´ is 0. -1 POINT each for
missing any one of them (up to -3 POINTS, of course).

d) 4 POINTS: there are two intervals where Q´ is positive. 1
POINT for each correct end point of an interval, correctly
presented. If an end point is included in an otherwise correct answer,
-1 POINT (taken off only once in this section of the problem).

**8. (12 POINTS)**

3 POINTS for writing a correct equation using the two formulas. 3
POINTS for writing a correct equation using the derivatives. 6 POINTS
for solving these equations, with 3 POINTS for getting correct
information about the intersection point and 3 POINTS for getting
correct information about B.