| Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | Total |
|---|---|---|---|---|---|---|---|---|---|
| Max grade | 16 | 12 | 12 | 10 | 12 | 12 | 12 | 14 | 95 | Min grade | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | Mean grade | 11.44 | 8.66 | 6.73 | 3.03 | 2.29 | 4.63 | 4.95 | 8.44 | 50.14 | Median grade | 13 | 10 | 7 | 2 | 1 | 3 | 8 | 9 | 51 |
Numerical grades will be retained for use in computing
the final letter grade in the course.
Here are approximate letter grade assignments for this exam:
| Letter equivalent | A | B+ | B | C+ | C | D | F |
|---|---|---|---|---|---|---|---|
| Range | [85,100] | [80,84] | [70,79] | [65,69] | [55,64] | [50,54] | [0,49] |
Minor errors (missing factor in a final answer, sign error, etc.) will be penalized minimally. Students whose errors materially simplify the problem will not be eligible for most of the problem's credit.
Problem 1 (12 points)
a) (6 points) 2 points for horizontal; 4 points for the others (1
point off for each element which is tilted up where it should be down
or vice versa, and 1 point off if the elements do not vary correctly,
more or less tilt, as scanned horizontally: max of 2 off for "tilt"
errors) and max of 2 off for sign errors.
b) (2 point) one equilibrium solution and specification of the
constant with the right equation (so x=CONSTANT loses a point because it does
not describe a solution to the differential equation).
c) (8 points) 1 point for correct separation; 3 points for integration
(1 of these is for a correct "+C"); 2 points for use of the initial
condition; 2 points for solving for y as a function of x.
Problem 2 (12 points)
Use of the arc length formula: 1 point; correct computation of
f´(x) earns 2 points; 2 points for the correct integrand;
2 points for the change to sec(x); 2 points for integrating secant;
2 points for evaluating sec and tan at the endpoints; 1 point for the
final answer recognizing ln(1).
Note This is essentially problem 10 in section 8.1 of the textbook.
Problem 3 (12 points)
a) (5 points) 2 points for the first and second derivatives; 2 points
for correct evaluation of the function value, and the first and second
derivatives; 1 point for final assembly of T2(x). ?????
b) (1 point) Earned for the answer. If the student gives a reasonable
polynomial as answer for a) this is earned by the value of the
student's polynomial at 5.
c) (6 points) The correct third derivative earns 2 points; estimation
of K in the correct way earns 2 points (selecting the correct endpoint
earns 1 of the 2 points but just selecting the incorrect endpoint
earns 0; the other point is earned for some reason why the correct
endpoint was selected); 2 points for putting everything together
(factorials, powers, etc.).
Problem 4 (10 points)
2 points for starting with a valid and relevant polynomial related to
ex; 2 points for correct substitution of -x2; 2
points for indicating multiplication of this polynomial by the
appropriate linear polynomial; 2 points for carrying out the
multiplication; 1 point for further algebraic expansion into standard
form (sum of constants times powers of x); 1 point for the answer. The
answer point is not earned if the polynomial does not have the
correct degree.
Problem 5 (12 points)
a) (6 points) Transforming the sequence to a form where
L'Hôpital's Rule could be used is worth 2 points; 1 point for
remarking on eligibility; 2 points for using L'Hôpital's Rule
(differentiation of the top and bottom); 1 point for the answer.
b) (6 points) 2 points for separating into two geometric series; 2
points each for the correct use of geometric series and the answer.
Problem 6 (12 points)
3 points for realizing or using the "key observation" that dropping
the square root increases the size of the fraction (using the other
part of the formula doesn't earn credit since it cannot be estimated
usefully); 3 points for showing that the infinite tail must be
estimated (here either a geometric series or a relevant [improper]
definite integral can be given); 3 points for estimating the infinite
tail by finding the sum of the relevant geometric series or with a
correct antiderivative; 3 points for using the sum and the
tabular information (or a correct integral!) to get a correct
answer. Merely asserting an answer is not sufficient to get credit
here (lots of data was supplied). Asserting that an infinite tail is
small because one or a few terms are small also earns no credit.
Problem 7 (12 points)
4 points for some evidence connecting the sum to the integral. One
acceptable item would be a picture similar to that displayed on the
answer sheet. An explicit inequality connecting the Nth
partial sum with a definite integral would also be acceptable. Also
useful would be mention of a relevant function decreasing. But
some evidence should be given.
4 points for evaluating a relevant definite integral. 4 points for a
correct answer with evidence showing that specific N is valid.
Students who use N in place of N+1 in an otherwise correct solution
will be penalized 2 points.
Problem 8 (14 points)
a) (8 points) 4 points for computing the ratio and correctly obtaining
a simple fraction (1 point is lost if the absolute value is missing);
3 points for obtaining the limit of the ratio; 1 point for the answer.
b) (6 points) 3 points for considering the case x=RIGHT END POINT. 1 of these 3 points is for
the answer, and 2 points for correct support of the answer.
3 points for considering the case x=LEFT END
POINT. 1 of these 3 points is for the answer, and 2 points
for correct supporting of the answer.