| 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. The statements of the questions should be a guide to whether an approximation is requested or allowed.
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 (8 POINTS)
1 POINT for y. 4 POINTS for the chain rule applied correctly. 3 POINTS for
the correct value of dy/dt. -1 POINT for not giving the exact value of one
or both trigonometric functions (the point is just taken off once!).
Problem 2 (10 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 part b); alternatively, some comment on
the behavior of the function must be given.
Problem 3 (12 POINTS)
3 POINTS for correct differentiation, 4 POINTS for finding the
critical numbers, 2 POINTS for saying what each one is, and 3 POINTS for
some explanation.
Problem 4 (14 POINTS)
a) 1 POINT.
b) 7 POINTS: 5 POINTS for correct use of the chain and product
rules. 2 POINTS for solving for the derivative correctly.
c) 5 POINTS: 2 POINTS 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!
d) 1 POINT.
5. (16 POINTS)
2 POINTS for the objective function (the area), and 2 POINTS for
writing it as a function of one variable using the equation. 4 POINTS
for differentiating the objective function correctly, 3 POINTS for
finding out where the derivative is 0, and 2 POINTS for explicitly
stating with identification what the sides of the rectangle with
largest area are. Finally, 3 POINTS for some explanation of why the
answer found provides a maximum (note that ANY explanation using
function values at endpoints or first derivative behavior or second
derivative value is fine but SOME EXPLANATION must be explicitly
given).
6. (20 POINTS)
a) 9 POINTS: 2 POINTS for computing W´(x) and 2 POINTS for computing
W´´(x). 3 POINTS for solving W´(x)=0 and 2 POINTS for
solving W´´(x)=0.
b) 1 POINT.
c) 10 POINTS: 2 POINTS for the graph and 3
POINTS for the labels and 5 POINTS for the answers. Note that the
domain of W(x) is x>0, and reference to intervals not in the domain
will be penalized by loss of 1 POINT.
7. (20 POINTS)
a) 8 POINTS: 0 POINTS for confusing the function and the derivative. 1
POINT for locating the correct x (+/- .25 accuracy is good enough). 3
POINTS for indicating it is a relative extremum. 4 POINTS for giving enough
explanation to conclude that the desired extreme value of the function
actually occurs at the number specified.
b) 12 POINTS: 3 POINTS for the linear approximation formula cited and
applied correctly. 2 POINTS for the derivative computed and evaluated
correctly. 1 POINT for the actual numerical answer. 3 POINTS for an
explanation of the discrepancy between the true answer and the
approximating answer involving the second derivative (these points can also
be earned for a picture of the function near the approximation together
with its tangent line). 3 POINTS for getting the necessary information
about the second derivative and its sign.