### The first 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 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.

### Discussion of the grading

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.

### The first exam

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:

 Letterequivalent Range A B+ B C+ C D F [85,100] [80,84] [70,79] [65,69] [55,64] [50,54] [0,49]

### Discussion of the grading

An answer sheet with answers to version A (the yellow cover sheet) is available, and here is a more compact version of this exam. The questions of version B were close to those of version A. I hope that students themselves will be able to create version B answers after reading the version A answers. 78 students took the exam. Statistical measures of the performance of the 39 students who took version A and the 39 students who took version B were quite close.

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.

### 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

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.

### Discussion of the grading

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.
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>

### The second exam

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

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:

 Letterequivalent Range A B+ B C+ C D F [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).

### Quiz for more points

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. Attached is an Excel file with the quiz scores. The quiz scores by section are in sheet 3 and the scores by version are in sheet 4.