### The first quiz on limits

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 5
Mean grade 4.75 2.32 4.61 3.25 4.34 3.87 23.14
Median grade 5 3 5 5 5 5 25

79 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". Each problem was worth 5 points.
The exam mostly tested algebraic skills which are essential for this course and in many other situations which use mathematics. Most of the problems were copied from textbook homework problems.

Students whose scores were 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)
Factor the top and cancel. Bad insertion or deletion of a minus sign in the algebra (after correct factoring!) and the answer earned a grade of 2 out 5: 0 for the answer and 2 out 4 for the supporting evidence.

Problem 2 (5 points)
I gave 3 points for substituting and getting the answer 0 if "plugging in" was presented clearly. I gave 5 points if in addition the word "continuous" was used correctly. I wanted some explicit reason here corresponding to the work needed to earn credit in other parts. If you got 3 out of 5 (which 50 out of 79 students earned in this problem!) maybe you could comfort yourself with the idea that, well, 28 out of 30 is a "good" grade.

Problem 3 (5 points)
2 points for factoring correctly but mistakes after that earned nothing more.

Problem 4 (5 points)
I looked for multiplication by "the conjugate". A sign error in the manipulation and answer was penalized 2 points (1 for the answer and 1 for an algebra error). Bad algebra with a correct answer will earn 1 point alone for the answer. I wanted correct supporting evidence!

Problem 5 (5 points)
I looked for the answers alone. The first answer (correctly given!) earned 1 point, and each of the next two earned 2 points each.

Problem 6 (5 points)
I looked for supporting algebraic evidence. Again, a sign error in the manipulation and the answer was penalized 2 points (1 for the answer and 1 for the algebra error).

### The first exam

Problem#1 #2 #3 #4 #5 #6 #7 #8 #9 Total
Max grade 12 6 14 12 12 8 12 14 12 99
Min grade 2 0 0 0 0 0 0 0 1 18
Mean grade 9.96 3.95 8.78 7.09 8.73 1.73 7.5 8.88 5.38 62
Median grade 11 5 9.5 6.5 10 1.5 10 9 5 65

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.

Students with grades of 50 or less
These students should carefully consider dropping this course. Students with very low grades in the first exam almost always cannot catch up with the material in the course. The first segment of the course spent substantial time reviewing ideas assumed to be known already. Almost all students who got D's or F's on this exam showed unsatisfactory understanding of background material and there will be no course time devoted to further review. Working on other courses where there is more chance of success may be an overall more productive strategy. Please discuss this with your academic advisors, and consider this recommendation seriously yourself.
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]

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. 82 students took the exam. Statistical measures of the performance of the 40 students who took version A and the 42 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 or Chain Rule lose 2 points.
Exception In c), an omitted multiplier of 3 (so (sec(3x))2·(cos(2x))2 beginning the quoted answer instead of (sec(3x))23·(cos(2x))2), although a Chain Rule error, will only lose 1 point.

Problem 2 (6 points)
2 point for the correct formula for the first derivative of sin(g(x)), 3 points for the correct formula for the second derivative of sin(g(x)), and 1 point for the correct answer, which must include evaluations of the trig functions. Again, 1 point off for a minor error and 2 points for an error in the Product or Chain Rule.

Problem 3 (14 points)
a) (10 points) 2 points for substitution of x+h into f (or something equivalent). If somehow the student writes 5-x+h instead of 5-(x+h) the student loses these 2 points, but they can be the only 2 points lost of the 10 if everything else is correctly done.
2 points for correctly instantiating [f(x+h)-f(x)]/h.
3 points for multiplying top and bottom by the conjugate and combining the fractions correctly.
2 points for cancelling the h's.
1 point for the answer, which must be the correct answer for the student's work.
2 points off if there is no mention of limh-->0 in an otherwise correct solution.
b) (4 points) The value of the derivative at 4 earns 1 point. A correct tangent line equation earns 1 point. As for the graph: a correct curve (the top half of a parabola opening to the left!) earns 1 point and a correct tangent line earns 1 point. The line must be tangent to the curve, or 1 point will be lost.

Problem 4 (12 points)
The correct derivative earns 4 points. Deduct points for differentiation errors here, but try to "read with" the student's result, if it is not trivial. For example, a student who begins by asserting that the derivative is 6x would (and should!) earn no points for the whole problem. Simplification to yield a correct equation with no quotient (so knowing that TOP=0 is enough) which finds where f´(x)=0 earns 4 points (2 points for f´(x)=0 and 2 more points for declaring that TOP=0). Deduct points for algebra errors here. The two coordinates correctly written earn 4 points. Some of that is deducted if incorrect "simplifications" are made.

Problem 5 (12 points)
3 points for s in terms of x (only 1 of 3 if there's some error in Pythagoras); 3 points for the area of the square; 3 points for the area of the triangle; 1 point for a correct formula for f(x); 2 points for a correct domain (with or without either or both endpoints).
Only 3 points are earned for a formula in terms of s and x.

Problem 6 (8 points)
The word continuous in relevant context earns 2 points. The phrase Intermediate Value Theorem in relevant context earns 2 points. This means that just scrawling (scrawl: "write in a hurried untidy way") the word "continuous" or the phrase "Intermediate Value Theorem" will not necessarily earn credit. 1 point is earned for a function value which is positive, and 1 point is earned for a function value which is negative. Some mention of the "size" of values of cosine should be made for 1 point. Finally, an explicit correct answer supported by the student's work earns 1 point.
Comment The problem statement contains this sentence: "Give careful evidence supporting your assertion, including specific results from Math 151." This instruction was interpreted strictly.

Problem 7 (12 points)
a) (7 points) Restricting attention to 0 and 2 is worth 1 point. 2 points for getting a constraint at 0. 2 points for a similar constraint at 2. 2 points for solving and reporting correct values of A and B.
b) (5 points) The graph should be continuous (1 point). There should be a straight line segment from (2,0) to (2,1/2) (2 points). There should be decreasing behavior as one moves to the left on the curve, but not lower than 1 (1 point). There should be decreasing behavior as one moves to the right on the curve, but not lower than 0 (1 point).

Problem 8 (14 points)
a) (10 points) The graph should have three discontinuities (1 point each), from left to right: one removable, one with a jump, and one with asymptotic behavior. 3 points for correct sign behavior in each interval (1 more point for f´(x)=0 for x>4). 2 points for correct {in|de}creasing behavior. 1 point for the asymptotic behavior.
While I think there should be a vertical asymptote at only one side of one discontinuity (as shown in my solution to this problem), I believe that a reasonable person might disagree with this, so I will not penalize answers which show another asymptote to the left of -1. I do not think that the behavior near the middle discontinuity of f´ is "asymptotic" -- that is, the left and right limits near the value are finite and have different signs.
b) (2 point) The correct answers.
c) (2 points) the correct answers.

Problem 9 (10 points)
a) (4 points) A correct answer earns 1 point. 3 points for combining the fractions, getting into simple fraction form, simplifying and plugging in.
b) A correct answer earns 1 point. Some analysis (or even just acknowledgment!) of absolute value earns 2 points, and then algebra gets the third point.
c) (2 points) A correct answer earns 1 point. Some support (the word "continuous"?) earns a second point.

### The second exam

Problem#1 #2 #3 #4 #5 #6 #7 #8 Total
Max grade 12 12 12 20 16 10 10 8 93
Min grade 0 0 0 0 0 0 0 0 11
Mean grade 8.49 3.52 8.72 8.46 10.85 4.83 5.24 1.75 51.86
Median grade 10 1 10 9 13 4 6 1 50

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.

Attendance
Students who answered at least 15 of the 18 QotD given before the second exam scored markedly higher (both mean and median) on the second exam than students who answered fewer than 15 of the QotD.

Comparison
The letter grades of 41 students went down from the first exam to the second exam (there really is much material in this course, even for students who may have studied calculus before). The letter grades of 21 students stayed the same (12 of those, however, were F/F combinations!). The letter grades of 9 students improved.

Here are approximate letter grade assignments for this exam (which are slightly different from the first exam):

 Letterequivalent Range A B+ B C+ C D F [80,100] [75,79] [65,74] [60,64] [50,59] [45,49] [0,44]

An answer sheet with answers to version A (the green 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. 71 students took the exam. Statistical measures of the performance of the 38 students who took version A and the 33 students who took version B were close. Wrong assumptions of linearity decreased the grades of many students in problems 2 and 8.

Problem 1 (12 points)
a) (1 point) Students should demonstrate knowledge that ln(1)=0 and show that the equation is correct at (-1,0). b) (9 points) Differentiating the equation earns 6 points: chain rule for ln (3 points), product rule for xy (2 points), all else (1 point). Slope at (-1,0) earns 1 point, going through (-1,0) earns 1 point, and a correct equation for the line earns 1 point. c) (2 points) The line should go through (-1,0) (1 point) and seem to be tangent (1 point).

Problem 2 (12 points)
2 points for labeling the variables and getting a function for the area; 3 points for differentiating the function correctly; 3 points for getting the roots when the derivative equals 0 (1 of these for setting the derivative equal to 0) and 1 point for rejecting one of the roots; 2 points for explaining why the max has been obtained; 1 point for the correct answer.
Computation of dy/dx if no connection with the problem is given earns 0 points.
Comment Many students assumed that three-sided shapes were triangles and somehow the appearance of several of these in the same picture implied that the "triangles" were similar. First, if only two of the sides are line segments, the result is NOT a triangle. Second, the non-triangular shapes are NOT similar: there is no reason to assume this is true.

Problem 3 (12 points)
a) (6 points) 1 point for an area formula, and 2 points for differentiating it. 1 point each for the answers.
b) (6 points) 1 point for a diagonal formula, and 2 points for differentiating it. 1 point each for the answers.

Problem 4 (20 points)
a) (6 points) 1 point for each of the limits. 4 points for justification (2 of these would be earned by just mentioning powers, with no detailed explanation carried out).
b) (5 points) 2 point for the answers. 3 points for justification. There must be use of the derivative.
c) (5 points) 1 point for the answer. 4 points for the justification, which should mention both {in|de}creasing to infinity, and some idea of continuity so that the line y=100 does intersect y=f(x). The student should give some idea why there are at least two roots of the equation given, and then some further idea why there are exactly two roots.
Comment I think this was the most subtle part of the exam. Although a number of students gave the correct answer, good supporting reasoning was given by only a few people. What's needed is a way of combining the limit statements and the derivative statements and some reason why the curve actually intersects the line.
d) (4 points) 2 points for the labeled inflection points on the graph (that's requested twice, once underlined), and 2 points for the concavity intervals (1 point if one interval is given correctly, and putting in/taking out endpoints doesn't matter here).

Problem 5 (16 points)
a) (9 points) 4 points for each vertical pair (x= and type) in the table. Both entries must be correct to earn the point. 5 points for convincing explanation (something resembling the First Derivative Test earns 2 of these and remarks about the specific signs of this derivative will earn the other 3).
b) (7 points) Each critical point correctly sketched earns 1 point. Going through (0,0) earns 1 point. A smooth curve defined on [-1,4] which resembles a good answer will earn 2 points.

Problem 6 (10 points)
a) (5 points) Correct locations of A1 and A2 with labels each 1 point each. A1 should be to the left of A and A2 should be to the left of A1, but to the right of 0. The assisting tangent line segments in each case earn another 1 point each. The limit earns 1 point.
b) (5 points) Correct locations of B1 and B2 with labels each 1 point each. B1 should be to the left of B and B2 should be to the right of B1 but to the left of -1. The assisting tangent line segments in each case earn another 1 point each. The limit earns 1 point.

Problem 7 (10 points)
a) (4 points) A correct answer earns 1 point. 3 points for checking L'H eligibility and differentiating (done twice). Be sure the second derivative on the "top" function should be correct or 1 point is lost.
b) A correct answer earns 1 point. 3 points for taking ln, rearranging, checking for L'H eligibility, differentiating, and exp back. Other arrangements (such as beginning by writing the function as eln(function)) can also be correct.
c) (2 points) A correct answer earns 1 point. Some support (the word "continuous"?) earns a second point.

Problem 8 (8 points)
The initial description of the model earns 3 points: a function and a domain. "Distance=Rate·Time" is not a description of a valid model. It is a recital of a formula which may or may not apply to this problem. The formula is valid for motion in a straight line with constant velocity and the problem statement does NOT state these conditions and therefore the formula cannot be used. The specific result needed is the Mean Value Theorem, and, in context, this will earn 2 points. Use of this result to get correct information earns the final 3 points, 1 of which is for a correct estimate. That 1 point can be earned with no supporting evidence.
Comment One student estimated (version A) that the distance between the towns ranged from 3600 miles (wider than the continental U.S.) to 5400 miles (more than the distance from New Jersey to Hawaii). Could this really be done by driving an hour and a half at a speed between 40 and 60 mph? On the other hand, another student asserted that the distance between the towns ranged from 6 to 9 miles, which also seems remarkable for a trip of an hour and a half with speed at least 40 mph.

### The final exam and course grades

Grading of the final exam was done on Tuesday, December 18, the day following the exam. Each lecturer and recitation instructor graded the same problems for all students. Certainly I should keep confidential which problems I graded, so all I will say is that I don't want to think more about trains leaving St. Louis, and I don't want to consider, at least for a while, piecewise defined functions which are to be both continuous and differentiable.

I spent the afternoon on the day after grading going through all of "our" exams in detail. Since few students will see their final exams (see more about this below) I felt a special responsibility to check the details. I found no mistakes in addition. I did discover many (well, maybe 6 or 7, so maybe not many!) discrepancies of 1 or 2 points in grading. That's to be expected in a big bunch of exams. I found two further errors, one unjustifiably penalizing a student 4 points, and another 11 point error. I'd feel very good about my checking all of the exams, except that the 4 point error was in a problem I graded, in my handwriting! What's this say about my own accuracy and concentration?

The results, overall and locally
The exam was taken by 617 students, and the results, on a 200 point scale, ranged from 0 (indeed) to 200 (yes). The overall median was 127. In our lecture, 65 students took the final exam. For this population, the grades ranged from 26 to 198, the median was 140, and the mean was 135.35. The letter distribution of grades on the final exam was definitely skewed higher for the students in sections 7, 8, and 9 than for the general population. The results showed that many students prepared diligently for the final exam. There was only one problem where "our" student solutions showed systematic weakness: the optimization question. When/if I teach 151 again, I'll try to improve my discussion of such problems (especially since the same weakness was displayed in the second exam). Final exam grades were converted from numbers to letters using the "bins" displayed below. The alert reader will note that these assignments are quite similar (multiply by 2) to those used earlier for the first and second exams.

 Letterequivalent Range A B+ B C+ C D F [170,200] [160,169] [145,159] [140,144] [120,139] [110,119] [0,109]

The final exam letter grades are given to allow individual lecturers, who are responsible for reporting course grades, to align the performance of their specific groups of students with the overall performance of students in the course. Random chance may give one lecturer a group of students with better or worse performance than the overall population, and, indeed, specific scheduling requirements may force specific subpopulations with different math preparation to enroll in some lectures. The common final exam and grading are part of the faculty's effort to assign appropriate course grades, with equal grades for equal achievement.