| Background about grading | ||||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Exam outcome | ||||||
Arithmetic errors will be penalized in the following way: -1 for the first error, and -1 for any additional errors. But students will need to follow the consequences - that is, they aren't allowed to just change their minds in the middle of a problem if their arithmetic errors have led to a more difficult situation to analyze than the correct one would have been!
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, 4 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.
Problem 1 (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
comment on the behavior of the function must be given.
Problem 2 (14 POINTS)
a) 2 POINTS for the statement of the definition of
f´(x) (leaving out "lim" in the definition loses 1 POINT!).
b) 8 POINTS for successfully manipulating the difference quotient and
getting the derivative. 2 POINTS of these are for inserting x+delta x
successfully in the formula for f. 0 POINTS for a correct answer which
is not supported by algebra.
c) 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-y0) DIVIDED by (x-x0) =
m) which is not satisfied by EVERY point on the line!
Problem 3 (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 parabolic curve, and 2 POINTS for each
correctly drawn line segment. If the parabolic curve bends
perceptibly, it should bend in the correct way, else -1 POINT. -2
POINTS for graphing several functions over the same interval.
Problem 4 (16 points)
a) 6 POINTS: a bald answer (with no justification) is acceptable
here. 2 POINTS for asserting that the length of BC is 5-x, and 2
POINTS more each for finding the area of each square.
b) 4 POINTS: the graph should be the graph of the function in a) and
should fit entirely within the "window" provided, and should never be
0. Such a graph should get 2 POINTS. If the graph goes markedly
outside of the window, -1 POINT. For the second 2 POINTS, the graph
should look like a parabola opening up, its minimum should be almost
halfway up, and it should be reasonably symmetric vertically (around
the line x=2.5).
c) 6 POINTS: 4 POINTS for showing some algebraic work (2 POINTS of
these for successfully "expanding" (x-5)2) and 1 POINT each
for finding valid x's.
Problem 5 (9 POINTS)
Each part is worth 3 POINTS. The answers alone can be written with
little effort. -1 POINT for a simple error (miswritten number, for
example). -2 POINTS for a product/chain/quotient rule error.
Problem 6 (8 POINTS)
2 POINTS for S(0), and 3 POINTS for each of S´(0) and
S´´(0). The numerical values of each of these are worth 1
POINT, so -1 POINT for any omitted value.
Problem 7 (19 POINTS)
The numerical answers in this problem will be read generously.
Student answers should be within about .5 of the answers given on the
answer sheet.
a) 5 POINTS: 1 POINT each. There should be five numbers given.
b) 4 POINTS: 1 POINT for each interval, consistent with the numbers in
part a). -1 POINT if an endpoint (NOT 2!) is included in one of the
intervals, and an additional -1 POINT if 2 is included in one of the
intervals.
c) 1 POINT for the answer.
d) 1 POINT for the answer.
e) 1 POINT for the answer.
f) 2 POINTS: 1 POINT each. There should be two numbers given.
g) 2 POINTS: 1 POINT each. There should be two numbers given, and the
answers should be consistent with the numbers in part f).
h) 3 POINTS: 1 POINT for each interval, consistent with the numbers in
part f).
POINTS can be taken off for more answers than are
correct. For example, the answer "All numbers" in h) will earn 0
POINTS.