Calculus 1 -- Fall 2017

Results for Exam 1

The class did an excellent job on exam 1.

The histogram below shows all of the scores (rounded to
the
nearest
whole number, with scores ending in .5 rounded up) for the first
exam. It
shows, for example, that one student earned the top score of 97, and
that five students (almost a third of the class) scored at or above
95. Almost
all of the students are on
track to earn a course grade of B^{-} or better at this point. You
can
find
your individual score posted on Blackboard. Contact
me if you need assistance accessing that. PLEASE review the score
posted for you on blackboard and confirm that it is the same as what
appears on your exam paper. If they are not the same, let me
know. Also, double check that your total for the exam has been
added up correctly. I am very careful totalling and recording
scores, but errors sometimes occur.

If you are dissatisfied with your performance on the exam,
please
make an appointment to see me. I may be able to suggest ways to
make your preparations more effective for the remaining exams.
And I am open to any suggestions about how I can make the course
more supportive of your efforts.

Grading Comments.

- In general, partial credit was awarded so that an A answer
would receive at least 90% of the points, a B answer at least 80%, and
so on. For any question on which you received less than 70% of
the credit, I am signaling an unsatisfactory response. With less
than 60% of the credit I am signalling a "failing" response.

- Multiple Choice. These items were marked treating each
option for each problem as a true/false item. One mark of x was
deducted for each item incorrectly circled, or incorrectly left
uncircled. Each x corresponds to a deduction of .75 on the
score. At the beginning of problem 1, the total number of x marks
and the total score for probs 1-5 is shown.

Although you can tell for each problem how many x's were marked, I deliberately did NOT indicate which items were correct or incorrect. In working on exam solutions, I want students to have something to think about on these items.

- In many cases students did not follow the directions in the
problem statement. For example, problem 6 calls for exact
answers, and for an indication of the rules being used. Most
students did not meet one or both of these requirements. On
problem 7, items asked about specific marked points, but several
students responded to one or more items by listing intervals. I did not deduct
much for such lapses on this exam, if I thought the answer demonstrated
understanding of the concept being tested, but don't count on that for
future exams. To maximize your score, read the directions
carefully, and follow the instructions.

- While grading the exams, I realized that problem 7g
involves a subtlety that we did not cover in class. Although the
graph is clearly concave up at point B
that does not necessarily imply that the second derivative is
positive. It is certainly the case that the second derivative
must be greater than or equal to 0, if it exists at all. But from
the given information, you cannot tell whether the second derivative is
positive, zero, or undefined. This can be seen in examples of the
form y = |x|
^{r}where r is close to 2. For r < 2 the second derivative does not exist at x = 0; for r = 2 the second derivative is 2 (so > 0) at x = 0; and for r > 2 the second derivative is zero at x = 0. But all three of these functions have a smooth u shaped graph with its bottom point at x = 0. You can verify these comments by graphing them on desmos, say with r = 1.8 or 2 or 2.2. Use the equations f (x) = |x|^{2.2}, y = f '(x), and y = f ''(x) to see graphs of the function and the first two derivatives, for example.

The fact is, I was sloppy in my thinking when I wrote the exam, and assumed (as almost all the students assumed) that the concavity at point B implies a positive second derivative. Full credit was given for that answer, because it is an easy error to make. But having discovered my error, I wanted to share it with the class. It helps remind us that calculus involves subtleties, and that we have to pay close attention to the details of the theorems and other conclusions stated in the text.