| Problem | #1 | #2 | #3 | #4 | #5 | #6 | Total |
|---|---|---|---|---|---|---|---|
| Max grade | 5 | 5 | 5 | 5 | 5 | 5 | 30 | Min grade | 0 | 0 | 0 | 0 | 0 | 0 | 1 | Mean grade | 1.55 | 3.79 | 4 | 2.12 | 1.55 | 3.31 | 16.31 | Median grade | 0 | 5 | 5 | 1 | 0 | 5 | 16 |
77 students took this quiz. It was the first "controlled" (limited time and help) test situation for these sections. Numerical grades will be retained for use in computing the final letter grade in the course. Each of the 6 problems was worth 5 points. The correct answer earned 1 point, and 4 points could be earned by the requested "supporting evidence". The exam mostly tested algebraic skills which are essential for this course and in many other situations which use mathematics. All of the problems used methods which were demonstrated in class, shown on the sample test, explained in the text, and needed to do the homework problems in the syllabus.
Students whose scores are less than 20 should be very concerned about their likely success in this course. Such students should ideally perceive their results as an alarm. Students should be spending at least 8 hours a week outside of class working on course material, and students with low scores should work on every suggested problem in the course syllabus. The course is relentlessly cumulative. "Catching up later" is practically impossible and students who think this are deceiving themselves.
Problem 1 (5 points)
Combine the fractions, convert from a compound to a simple fraction,
and then factor and cancel.
Problem 2 (5 points)
"Plug in". Since the denominator is non-zero, the function is
continuous at 9, and the limit is the function value at 9.
Problem 3 (5 points)
Divide top and bottom by x3 and consider the asymptotic
behavior of all the parts of the expression.
Problem 4 (5 points)
Multiply by the conjugate and cancel, or just factor x-9 and then
cancel.
Problem 5 (5 points)
The top --> a negative constant, and the bottom is small and
positive. The result is negative and large.
Problem 6 (5 points)
Expand the squares and cancel x2's. Then factor and cancel.
| Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | Total |
|---|---|---|---|---|---|---|---|---|---|
| Max grade | 12 | 5 | 15 | 14 | 16 | 8 | 16 | 14 | 95 | Min grade | 0 | 0 | 0 | 1 | 3 | 0 | 0 | 0 | 15 | Mean grade | 10.92 | 3.40 | 8.15 | 10.17 | 11.45 | 3.74 | 8.69 | 8.92 | 65.45 | Median grade | 11 | 4 | 7 | 11 | 12 | 4 | 8 | 9 | 67.5 |
Numerical grades will be retained for use in computing the final letter grade in the course. Students with grades of D or F on this exam should be very concerned about their likely success in this 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] |
Problem 1 (12 points)
Each part is worth 4 points. Full credit is earned by the answer
alone. Minor errors (such as labeling 5·3 as 12, for example,
or a sign error) will be penalized 1 point. Errors in the product or
quotient rule lose 2 points.
Problem 2 (5 points)
1 point for the correct value of F(2).
1 point for the correct value of F´(2), and 3 points for showing a
valid process. Again, 1 point for a minor error and 2 points for an
error in the product rule.
Problem 3 (15 points)
a) (2 points) 1 point for a correct algebraic formula, and 1 point
for a correct limit.
b) (13 points) 2 points for substitution of x+h into f (or something
equivalent). 2 points for "expansion" of f(x+h). 2 points for
combining fractions correctly. 2 points for cancelling +/- objects,
and 2 points for cancelling a multiplicative h. 2 points for the limit
(some statement, even "-->", otherwise 1 point deducted). 1 point for
stating the answer.
The answer alone or the answer obtained algorithmically earns no credit.
Problem 4 (14 points)
a) (11 points) The graph should have two discontinuities (1 point
each), one with a jump. Elsewhere the graph should be nice and
continuous (2 points), and the "jump" behavior if not shown will lose
1 point. The graph should be 0 at two points (1 point each) and one
interval (1 point) and have the correct regions of positivity and
negativity (4 points or 1 point each). I think there should be a
vertical asymptote at only one side of one discontinuity (as shown in
my solution to this problem)but I believe that a reasonable person
might disagree with this, so I will not penalize answers which show
another asymptote or which show another jump: but a jump or asymptotic
behavior should be shown at each discontinuity of f´(x) (these
certainly are not removable discontinuities of the derivative!).
b) (1 point) The correct answer.
c) (2 points) the correct answers.
Problem 5 (16 points)
Each part is worth 4 points, and an unsupported correct
answer in each part earns 1 of these points.
a) Factor and divide.
b) Division by a power of x or some reasoning.
c) "Plug in" and mention that the bottom (the denominator) is not 0 or
that the function is continuous at 0. If not mentioned, 1 point
penalty.
d) Some analysis of the sign and size of the bottom is needed.
Problem 6 (8 points)
a) (2 points) 1 point for some correct values of K and L, and 1 point
for support of this assertion.
b) (3 points) 1 point each for the {positive|negative} answers, and 1
point for support of the assertions.
c) (3 points) 1 point for a citation of the Intermediate Value
Theorem, 1 point for a correct use of the word "continuous" in
connection with this function and citation, and 1 point for an
appropriate interval in the (x) domain variable.
Problem 7 (16 points)
2 points each (total of 4) for connecting the values of f(1) and
f´(1) with the given equation of the line.
4 points for correct differentiation of f(x) (including the A and B
appropriately!).
2 points each (total of 4) for f(1) and f´(1) in terms of A and
B.
Now 2 points for the equations connecting the two "views" of
f(1) and f´(1).
2 points for the correct answers.
Problem 8 (14 points)
a) The function earns 6 points. The domain earns 2 points (with or
without either or both endpoints).
b) The function earns 6 points. 2 points will be earned for each
formula and 1 point for for each correct simple specification
("simple" here means a restriction on x alone written as an
inequality, so x<17 would be such a specification but x+7<3x+4
would not). The correct formulas interchanged with respect to correct
specifications will earn only 2 of 4 formula points.
70 students took this quiz. Numerical grades
will be retained for use in computing the final letter grade in the
course. Each of the 6 problems was worth 5 points.
Results on this quiz were generally good. Generally, 2 points were
deduced for "major" errors in using, say, the {Product|Quotient|Chain}
Rules, and 1 point was deducted for more minor mistakes.
Numerical grades (the New total, which is the sum of the exam
grade and the grade on the quiz, please see below) will be retained
for use in computing the final letter grade in the course. Here are
approximate letter grade assignments for this exam:
The second computational quiz
Problem #1
#2
#3
#4
#5
#6
Total Max grade
5
5
5
5
5
5
30
Min grade
0
0
0
0
0
0
8
Mean grade
4.6
4.3
3.2
3.9
4.4
3.8
24.3
Median grade
5
5
4
4
5
5
26.3
Discussion of the grading
Some students had difficulty applying the Chain Rule (when to use it
and how to use it correctly). Also, please note that arctan(x) is
not the same as 1/tan(x). Arctan is the function which is
inverse to tangent, and arctan has carefully selected domain
and range. Additionally, there were some difficulties with implicit
differentiation. Here the advice would be to stay calm, and do the
algebra correctly.
p>
Problem #1
#2
#3
#4
#5
#6
#7
#8
Total
BPQ
New total
Max grade
12
12
12
12
12
12
22
6
94
28
122
Min grade
0
0
0
0
0
0
0
0
15
0
23
Mean grade
6.09
9.05
5.49
5.12
4.12
3.97
10.27
1.68
45.8
19.34
65.15
Median grade
6
10
5
5
3
4
11
1
44
20
64
Letter
equivalentA B+ B C+
C D F Range [85,100] [80,84] [70,79]
[65,69] [55,64] [50,54] [0,49] Discussion of the grading
An answer sheet is available, and here is a more compact version
of this exam. 75 students took the exam.
Due to the generally poor grades recorded on the exam, an opportunity was given for students to earn more points. A bonus points quiz was given (answers here) with some grading discussion here. 73 students took the quiz.
Problem 1 (12 points)
1 point for differentiation of the function; 2 points for factoring the
derivative; 3 points for spcifying the critical points; 4 points for
the needed values of f; 1 point each for the maximum and minimum
values of f.
Problem 2 (12 points)
a) (1 point) Plug in the numbers and check.
b) (6 points) From left to right in the original equation: Chain Rule
(1 point); derivative of x3 and 3 (1 point); Product Rule
(1 point), and 1 point for dy/dx appearing correctly in both places.
2 points for getting a formula for dy/dx. The 2 points for solving for
dy/dx can be earned even if there is a mistake in differentiation but
only if dy/dx appears twice in the student's previous computation and
the successive algebraic manipulations are correct. Also, the 2 points
can only be earned for "uninstantiated" dy/dx: that is, no
substitutions for x or y have been made (the problem does specifically
request an answer "in terms of x and y").
c) (3 points) 1 point for realizing that the line must go through
(-2,1), 1 point for the slope, and 1 point for a valid equation of the
line.
d) (2 points) The line should go through (-2,1) (1 point) and seem to
be tangent (not cross the curve at the point of tangency!). The
direction should be correct. (1 point)
Problem 3 (12 points)
a) (4 points) 2 points for writing tan(theta)=A/B, and 2 more points
for rewriting as theta=arctan(A/B).
b) (8 points) 1 point for writing the needed value of theta as
arctan(10/5) (or 2!). 5 points for computing d(theta)/dt using the
Chain Rule and then the Quotient Rule. 2 points for evaluating
d(theta)/dt at the desired instant by inserting the supplied
numbers. Students who use the equation tan(theta)=A/B can still earn 5
points for differentiation by differentiation the equation they have
implicitly. More points can be earned by inserting the supplied
numbers. If no correct value of theta are given and only the supplied
values of A, B, and their derivatives are correctly used, 1 point will
be earned.
Problem 4 (12 points )
a) (7 points) The value of A(1) is worth 1 point. Formula for A'(x) is
4 points (2 points for chain rule and 2 points for any further work
towards the answer [plugging in C(x)]). The value of A'(1) is worth 2
points.
b) (3 points) Linear approximation formula is 2 points, with 1 point
(which may be lost if incorrect arithmetic is then used!) for
instantiation of the formula. Students may use either the correct
values or the values they have computed in a).
c) (2 points) {Over|under} correctly answered is 1 point. The reason
is worth 1 point.
Problem 5 (12 points)
3 points for the preliminaries: draw a picture, label the picture
(including the sides of the rectangle -- that's 1 of the 3 points),
and write an area formula in terms of the sides of the rectangle.
2 points for finding a connection between the sides of the rectangle
by using the geometry of the triangle or otherwise.
2 points for writing the area of the rectangle in terms of one
variable.
2 points for the derivative of the rectangle's area and finding the
critical point.
2 points for the statement of the solution (the area and the
dimensions). 1 point for some explanation of why the answer is a
maximum.
Problem 6 (12 points)
Each part is worth 4 points: 1 point for the actual (correct!) answer,
and 3 points for some explanation. The comments refer to what's needed
for the explanation.
a) Use l'H twice, mentioning eligibility each time (or show evidence
that some check of this has been done).
b) Take logs, use l'H and mention eligibility, then exponentiate the
result.
c) "Plug in": that is, know the behavior of arctan and exp when x is
large.
Problem 7 (22 points)
a) (2 points) 1 point for the answers and 1 point for some supporting
evidence.
b) (2 points) Correct differentiation.
c) (2 points) 1 point for the coordinates of each critical point.
d) (3 points) 1 point for each interval (the endpoints won't matter
here).
e) (6 points) This part was the most difficult to grade. Of course what's
wanted is a graph which is correct. But what's sketched should be
consistent with the student's evidence in a), c), and d).
f1) (1 point) For an answer which is correct and supported
by other evidence in e) or is consistent and clearly supported by the
student's graphical answer in e). An unsupported answer (a "guess")
will not receive credit.
f2) (1 point) For an answer which is correct and supported
by other evidence in e) or is consistent and clearly supported by the
student's graphical answer in e). An unsupported answer (a "guess")
will not receive credit.
g) (2 points) The answer should be correct and supported by other
evidence such as the graph in e) or should be clearly and
non-trivially supported by the answer in e) together with limit
evidence in a). 0 or 1 point may be earned if the answer, although
consistent with the student's evidence, has become much easier than
the correct answer (for example, if one or both endpoints involve
infinity)
h) (3 points) 1 point for the number of inflection points and 2 points
for their location. This will be graded similar to the previous three
sections of the problem. Guesses unsupported by evidence will earn no
credit. Again, if the student's graph unduly simplifies the problem,
full credit cannot be earned here.
Problem 8 (6 points)
a) (3 points) 1 point for citing the Mean Value Theorem, 1 point for
mentioning the differentiability of the function, and 1 point for the
desired conclusion.
b) (3 points) 1 point for citing the Intermediate Value Theorem, 1
point for mentioning continuity of the function, and 1 point for using
the "evidence" (the differing signs of the function).
| Problem | #1 | #2 | #3 | #4 | #5 | Total (for BPQ) |
|---|---|---|---|---|---|---|
| Max grade | 6 | 6 | 3 | 8 | 7 | 28 | Min grade | 0 | 1 | 0 | 0 | 0 | 6 | Mean grade | 4.75 | 5 | 2.27 | 2.71 | 5 | 19.74 | Median grade | 5 | 6 | 3 | 1 | 5 | 20 |
The numbers above refer to those students who took the quiz. The
BPQ numbers are different, since more students took the second
exam than took the quiz. The quiz grading was done by Ms. Blight, and
here are some remarks.
Problem 1 (6 points)
Remember the product rule. Do algebra correctly.
Problem 2 (6 points)
Please, again, read the problem, and answer the questions which are asked.
Problem 3 (3 points)
Sign matters.
Problem 4 (8 points)
The rectangle was displayed, and its width is 2x, not x.
Also, some reason should be supplied to support the assertion that a
rectangle of largest area has been found, rather than smallest
or even neither (an inflection point).
Problem 5 (7 points)
Students should know simple values of trig functions. Again, sign matters!
Maintained by greenfie@math.rutgers.edu and last modified 11/28/2006.