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 6 and 7. The statements of the questions should be a guide to that.
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.
Partial credit was assigned according to the recipes below. Some of the instructors in this course may allocate partial credit more strictly than what's indicated here.
2. (16 POINTS)
a) 3 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 "cubic" curve, and 2 POINTS for the line segment. -2
points for graphing both functions over the entire interval.
c) 4 POINTS: 2 POINTS for each domain & formula. The assignment of x=1
could be done in either "piece".
d) 4 POINTS: this should be a reflection across y=x of the answer to
b), otherwise -2 POINTS. As in b), the graph should be continuous,
with the same assignment of point values for each piece.
3. (20 POINTS)
I think these questions were all copied from assigned homework
problems! 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 part d).
4. (10 POINTS)
The first part of the problem is worth 4 POINTS. Deductions: 1 POINT
off for inverting the fraction, 2 POINTS off for writing tan instead
of arctan, and 2 POINTS off for incorrect cancellations in and out of
the functions. The second part of the problem is worth 1 POINT. The
last part of the problem is worth 5 POINTS. Again, 1 POINT off for
inverting a fraction incorrectly and 2 POINTS off for incorrect
cancellations in and out of the functions. 2 POINTS for writing a
correct equation for this problem involving sin.
5. (16 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) 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.
c) 5 POINTS: 1 POINT each for the requested labels. 1 POINT for the
straight line, 1 POINT for the curve, and 1 POINT for their tangency
(they should touch "tangently"!) at the correct point of the graph.
6. (12 POINTS)
Certainly for a) and c) the answers alone can be
written with little effort: 3 POINTS for each of these. Part b) is
worth 4 POINTS, and part d) is worth 2 POINTS, one for each value.
7. (10 POINTS)
a) 2 POINTS: 1 POINT for drawing the tangent line, and
1 POINT for the numerical answer.
b) 8 POINTS: 2 POINTS for f'(x). 2 POINTS for solving where f'(x)=0
and getting the correct answer, 1 POINT for plugging in f(ln 3), 1
POINT for expressing the answer suitably, and, finally, 2 POINTS for
stating what the coordinates of A are.