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\centerline{Mathematics 131, section 1 -- Calculus for Physical and Life Sciences}
\centerline{Discussion 1 -- Derivative Functions}
\centerline{\it September 29, 2004}
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{\it Working in a Group}

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Since this is the first of the discussion class 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 {\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.  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, to get the most out of this kind of work, 
some of 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.  
\item{$\bullet$} If you feel totally``clueless'' at some point
and everyone else seems to be ``getting it,'' 
your job will be 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!)
\item{$\bullet$}  On the other hand, when you think you do see 
something, you need to be willing to explain it patiently to 
others.  (Don't forget, the 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.)   
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In short, {\it everyone has something to contribute, and 
everyone will contribute in different ways at different times}.
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{\it Discussion Work}
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The goal today is to generate reasonably accurate sketches of
the graphs of the {\it derivative functions}
$y = f'(x)$ for the functions $y=f(x)$ given
in problems 2,3,5,7 from Section 2.4 in the text book. 
``Reasonably accurate'' here means you should be aiming for
{\it more than} just a qualitative picture of $y = f'(x)$ along
the lines of what we did in class yesterday.  

How can you improve on that?  Well, note that the graphs 
are printed {\it on a grid}.  The small squares in the grid
are {\it one unit on a side} in each case.  If you draw in 
an approximate tangent line to the graph (``eyeball it'', but 
carefully), at an integer $x$ value, then you can estimate
the slope of that tangent line using the grid.  That slope
will become the $y$-value for the derivative function!
 
Turn in your group's four graphs of the derivative functions
on sheet(s) of graph paper provided.
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{\it Suggestions:}  
\item{$\bullet$}  Don't mark the additional
sheet until you have worked out what the graph $y = f'(x)$
should be to the satisfaction of all group members
and you can make a ``clean'' graph without erasing,
redrawing, etc.  
\item{$\bullet$} Since you may need to redraw in 
any case, use a pencil for this work, not a pen(!)
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{\it Assignment}
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Group graph sheets due {\it at the end of class}.
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