20F test 1 remarks

All three problems were taken right from the textbook exercises.

Many students are still forgetting that the correct notation for an integral includes the ending differential of the integrating variable. Using proper mathematical notation is part of understanding the material and communicating unambiguously. I will begin taking off points for these kinds of obvious notational errors in the future.

In any of the plots required, many students made an ineffective sketch often without axis labels, axis tickmarks to indicate numerical values, not actually drawn carefully from a technology plot. Many graphs were tiny, often with a large part of the "field of view" empty. For problem 2 it was explicitly requested that only t >0 be shown, so if some negative values were shown for t, the rapidly growing negative exponential squashed the meaningful part that needed to be shown, often with a large empty region above that as well. Part of graphing is understanding how to best display the numerical values taking full advantage of a nice rectangular plot window. Transferring a good plot from a Maple window using its grid to locate key points and then tracing in a curve is the best way of making such a hand sketch. Otherwise, print your plot out and hand annotate it.

1. Logistic population growth. standard.
   Word problem, so units are important. Time units are months. Many people never thought about converting 5 years to months.
   With a small number of population, the discreteness of the approximation becomes evident, so one should take it with a grain of salt, as for example the one percent of the stable population is a fractional cow.
   
   When you plot the S curve, remember that when it crosses the halfway mark to the stable population, it should change concavity in your hand sketch. For those of you who made remarks about when the population itself is one percent of the stable population, this is a very negative time to the left of the rest of the S curve to the left of the vertical axis t = 0. "within one percent" means to reach .99 times the stable population.
   
   Although I continually emphasized the point, that all arbitrary constands in a solution derivation should be distinguished, espectially when it is convenient to use the implicit solution to evaluate the constant instead of resolving the final form of the expression in the explicit solution. Yet, many use the same letter C for distinct constants.
   
   
2. Linear DE. This is a purely mathematical problem, so any numbers should be calculated exactly before approximating to a decimal number. The first line of the instructions are "Show all work, including mental steps, in a clearly organized way that speaks for itself." That means that each step in the linear solution algorithm should be explicit. No steps jumped over. Finally "Always simplify expressions." Many left the solution as a product of one exponential with a sum of two terms, one of which is an exponential. Only by multiplying through does one get a simple expression which is trivial to both differentiate and integrate, but the product function from which it comes cannot be directly integrated or differentiately easily without first multiplying out, hence the simplest form is multiplied out.
   
   Similarly a constant divided by an exponential must first be converted to a sign-reversed exponential before either integrating or differentiating, hence is simpler after converting it.
   
   The critical point should be found exactly, then approximated.
   
   
3. Separable DE. Deceleration. The parenthetical remark at the end indicates why this problem is interesting, that for a 3/2 power function deceleration parameter, the total coasting distance is finite, so if we put numbers to the problem, clearly this is the interesting number to evaluate. "How far does the body coast?" clearly refers to this distance. Key formulas just had to be derived so the goal was clear. Forcing the derivation to reach this result while being sloppy about algebra and ignoring sign problems that were a clear signal of a previous error is now how one does math.
   
   "What is the limiting displacement (change in position) as t approaches infinity?"
   Displacement is a change in position (emphasized in the parenthetical remark to help you not make this mistake), yet many responded only with the final location, not the change in location. READ the words. Artificial math problems with no words and no context are useless for developing the skills needed to use math in applied fields, which is why all of you are in this course.
   
   "How long does it take for the body to reach within 1 ft of the stopping location? " Yet many responded using the location at t = 1/2 when it takes an infinite time to reach the finite stopping location.
   
   One can learn a lot more from this final problem, including a time scale for reaching the final destination. If interested read this:
   threehalvespowerdeceleration.pdf
   
   

Submission problems.
In the future, I will not grade poor test scans which are hard to read on screen or which are not single PDF files.

Use black ink (not blue) to do your work and don't bunch it all up together. Space it out going down the page, putting derived results at the END of the derivation, not circling back and putting them at the beginning. Box only the quantities actually requested in each part. For graphs, take them seriously and make them detailed, labeling everything important for the problem and including tickmarks etc.

Use a good scan app like Adobe Scan which compensates for poor lighting, and autoselects the border of the page and then makes the scan into a standard 8.5x11inch rectangle scan, with all pages in ONE PDF with filename
    LastName-FirstName-Test2.pdf (next time)
I have to extract the files to a folder to grade them so the filename is key to alphabetizing them for uploading again and recording grades.