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\centerline{Mathematics 36, section 5 -- AP Calculus}
\centerline{Discussion 1 -- Using Derivatives}
\centerline{September 22, 1999}
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{\bf Working in a Group}
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Since this is also the first of the discussion classes of the semester,
a few words about this way of working are probably in order.
In the discussion meetings of this class, we will be aiming
for truly {\it collaborative learning\/} -- that is, for an
integrated group 
effort in analyzing and attacking the discussion questions.
The ideal is for everyone in each of the groups to be fully
involved in the process.  The idea is that, by actively 
participating in the class through talking about the ideas
yourself in your own words, you can come to a better first 
understanding of what is going on than if you simply listen 
to someone else (even me!) talk about it.  

However, it must be said that to get the
most out of this kind of work, you may have to adjust some
of your preconceptions.  In particular:
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\item{$\bullet$}  This is {\it not a competition\/}.  
You and your fellow group members are working as a
team, and the goal is to have everyone understand what the 
group does fully.

\item{$\bullet$}  At different times, it is inevitable that
different people within the group will have a more 
complete grasp of what you are working on and
others will have a less complete grasp.  Dealing with
this a group setting is excellent preparation for real
work in a team; it also offers opportunities for significant
educational experiences:  

\itemitem{a)} If you feel totally ``clueless'' at some point
and everyone else seems to be ``getting it,'' 
you need to feel free to ask questions
and even pester your fellow group members until the point
has been explained to your full satisfaction.  (Don't forget,
the others may be jumping to unwarranted conclusions, and
your questions may save the group from pursuing an 
erroneous train of thought!)
\itemitem{b)} On the other hand, when you think you do see 
something, you may need to be willing to explain it carefully to others.
(Don't forget, the {\it absolutely best\/} way to make
sure you really understand something is to try to 
explain it to someone else.  If you are skipping over an
important point in your thinking, it can become very apparent
when you set out to convey your ideas to a team member.)   

\item{} In short, everyone has something to contribute, and 
everyone will contribute in different ways at different times.
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{\bf Background}
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Our techniques for analyzing {\it critical points\/} of functions
can be applied to many problems, especially optimization problems, 
where we seek a maximum or minimum value of some function.

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(Over for questions)
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{\bf Discussion Questions}
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\item{A)}  The following plot shows the graph $y = f'(x)$ 
(the {\it derivative})  for some function $f$.
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item{}Using this information, sketch a qualitative plot first for $f''(x)$.
Then use all the information you have to sketch $y = f(x)$, assuming
$f(0) = 1$.  Say where $f$ is increasing/decreasing, concave up/down,
and give the locations of all local max and min and inflection points.

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\item{B)}  To give a patient an antibiotic slowly, the drug is injected
into a muscle (rather than directly into a vein).  The quantity
of the drug in the bloodstream of the patient starts at 0 at $t = 0$
when the injection is given,
increases to a maximum, then decays asymptotically to zero.  
The rate of increase of the quantity of the drug is greatest at the 
time of the injection.
\itemitem{1)}  Sketch a possible graph for the quantity of the drug
in the bloodstream as a function of time.
\itemitem{2)}  Using that graph, sketch a graph of 
the {\it rate of change of the amount of the drug\/} -- the 
rate at which the drug is entering or leaving the bloodstream. 
\itemitem{3)}  Construct a third graph showing the rate of change of 
the rate of change of the quantity of the drug.  What is the meaning 
of the $t$-axis intercept of this graph? 
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\item{C)}  The efficiency of a screw is given by 
$$E = {(\theta - \mu\theta^2)\over \mu + \theta}$$
where $\theta$ is the pitch angle of the threads of the screw, 
and $\mu$ is a positive constant.  What value of $\theta$ 
maximizes $E$?

\item{D)}  You have a 10 inch length of flexible metal wire.  If
you cut the wire into two pieces, bend one to enclose a square, and
the other to enclose a circle, what is the {\it largest\/} total
area you can enclose?  Give a full justification of your answer.

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{\bf Assignment:} 
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One write-up per group 
of solutions for these problems, due Monday, September 27.
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