Comments on the first exam

Scores for the first exam have been posted on blackboard. Exams will be returned in class on Friday, 9/30/16. It is important that you come to class to pick up your exam because you are assigned to work with a group to produce a complete set of correct solutions that will be due Friday, 10/7. Exam solutions are a required component for the course portfolio. Specific directions are given here.

When you get your exam, please double check that I have added up the points correctly and also that the score on your paper is the same as the one posted on blackboard. I try hard not to make errors totaling or recording scores. But if an error has been made, I will want to correct it as soon as possible.

The exam scores are shown in the histogram below.

The histogram shows that most of the class did very well, with nearly 2/3 of the scores are at 90 or above. At this point everyone is earning a satisfactory grade, and almost everyone is on track for a final grade of B or better.

Anyone who is unhappy with his or her performance should have a discussion with me. I may be able to suggest ways to study more effectively.

Partial Credit System. I assign partial credit on each part of each problem by thinking about how I would assign a letter grade to that particular part. If I think an answer deserves an A I assign be 90% and 100% of the points; if I think it deserves a B

I assign between 80% and 90% of the points. So if you got less than 70% of the points, that means I thought that particular answer was unsatisfactory. If you got less than 60% that means I thought that particular answer was worthy of a failing grade. This tends to make my partial credit assignments somewhat more generous than many other math teachers. On the other hand, I rarely curve an exam. If you got a total for this exam in the 90's, then your overall grade for the exam is in the A range.

**Portfolios.
**If you handed in a portfolio for review, please look at the review
sheet I put in the front. If I requested any changes to what you
have put together, please make those changes as soon as
possible, and before the next exam. The review sheet must stay in
the very front of your portfolio. If you wanted to have your
portfolio reviewed but forgot to bring it to the exam, you may bring it
to my office hours by Thursday, 10/6 and I will review it
there while you wait.

Comments About Specific Exam Questions

Problem 1. I counted two kinds of errors: circling a term that does not apply, or failing to circle one that does apply. For each part A, B, and C, I indicated the number of errors like this: 2X means two errors. But I did not specifically indicate where the errors are -- you will have to figure that part out.

Problem 2. The class as a whole did not do very well on these discussion items. This indicates to me that most students are missing some key concepts about differential equations. I will summarize key points below

2A. Here is the key idea of an
autonomous
model. If you perform two experiments with the same initial
conditions but with different starting times, do you observe the same
behavior of the solution curve? If so, then the model is
authonomous. Matheamtatically, this means that the later solution
curve will appear to be the same as the earlier solution curve, but
shifted horizontally consistent with the later starting time. For
algae growing in a pond, we would NOT expect this to hold. Algae
need sunlight to grow, so if you started two experiments with the same
initial conditions but starting one at sunset and the other at sunrise,
you would predict quite different population curves to result. So
we would not expect an autonomous model to be valid in this case.

2B. The key ideas are
as follows.

1.
A
solution going to infinity in finite time
means that on a particular solution curve there is a vertical asymptote for some *t**,
or in other words, on that curve y
goes to infinity as *t* approaches *t**.

2.
The
practical significance is that from an initial state, the model only
allows us
to predict future evolution for limited amount of time.
In the case of the example, for any initial
point (say (160,29)) there is a unique solution curve (*y*
= tan(*t* + *C*) where *C*
= arctan(29)-160). We can
predict what *y* will be for a short
amount of time only. That is, given that
*y* = 29 at time 160, we can predict
what *y* will be at time 161, but not
at time 163. For a model of a real
system, this might indicate a flaw in the model (if we have reason to
believe
the system will continue for extended periods of time, or that physical
constraints impose a limitation on the size of the y variable) but it might
also
predict some catastrophic event in the future of the model. In a population model, for example, it is
possible for the model to predict a population that reaches 0 in finite
time. In practical terms, that means the
population has died out. The model might
predict that the solution curve continues into negative P values beyond
that
point, or even goes to negative infinity in finite time.
These don’t necessarily invalidate the model
– we know that the only relevant part of the model is where it predicts
positive population values.* *

3.
The
final question asks whether there are any conditions that tells us that
solution curves do not go to infinity in finite time.
This is not about the specific example given,
but for differential equations in general.
If you are given a differential equation
and an initial condition, is there a way to predict whether the
solution
curve might go to infinity in finite time?
Here is one answer: for an autonomous equation, look at the
phase
line. If the conditions of the existence
and uniqueness theorems hold, and if the initial *y *value
is lower than an equilibrium solution, then the entire
solution curve must be below that equilibrium value, and therefore *y* cannot go to infinity.

2C. The key ideas are as follows.

1. The student’s conclusions are exactly the sort we would derive from a qualitative analysis and phase line for the given problem.

2.
And
yet, if you solve the equation (it is separable) you will find that the
conclusions are incorrect: solution curves that begin above the *t* axis do not approach that axis as an
asymptote.

3.
How
can this be? The answer is this: our
qualitative analysis methods are valid when uniqueness holds. (Why is that?
How do we use uniqueness to derive conclusions about asymptotes?)

4.
A
complete correct answer for your exam solutions should explain where
and why the uniqueness
theorem
does not apply for this problem, and also explain specifically why this
failure
invalidates our usual conclusions about asymptotes.
It should also find the solution curves for
the given differential equation, and show directly that the conclusion
about
asymptotes is incorrect.