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\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Discussion 1 -- Linear and Exponential Functions, continued}
\centerline{September 7, 2005}
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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 where everyone in each of the groups is
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, 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 (the way many desirable jobs in the 
real world are structured); 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 get your fellow 
group members to explain what they are saying 
until  you are fully satisfied.  (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 {\it really understand} something is to be able 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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\noindent
In short, {\it  everyone has something to contribute, and 
everyone will contribute in different ways at different times}.

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\noindent
{\it   Background}
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We have seen that linear functions can be described by equations
$y = mx + b$ or $y = m(x - x_1) + y_1$.
Exponential functions, on the other hand look like
$y = ba^x$. In the first question, we will explore some of the information that
is contained in the equation, and how that relates to the graphs
of linear functions.
Finally question D deals with the difference between
exponential and linear functions.  (Turn over for questions).


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{\it  Discussion Questions}

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\noindent
A) (No graphing calculators on this question!) Do problem 11 
from Section 1.1 (page 7) of the text.
Give full explanations of how you are matching the formulas with the 
graphs, in one or more complete sentences.

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\noindent
B)  Recall that saying a quantity $y$ is {\it  (directly) 
proportional} to another quantity $x$ is the same as saying 
$y = kx$ for some constant $k$.  
\item{1)} Explain why saying that $y$ is {\it  (directly) 
proportional} to quantity $x$ says that $y$
is a linear function of $x$.
\item{2)} For example, the circumference of a circle is 
proportional to the radius of the circle.  What is the constant 
of proportionality in that case?  
\item{3)}  Write as a formula:  "the energy $E$ expended by 
a swimming dolphin is proportional to the cube of its speed $v$."

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C)  One potentially tricky point in section 1.2 in our text
is the distinction between growth rates and {\it  continuous} 
growth rates.  Here's the ``run-down'':  Saying $P$ grows or decays at a
rate of $r\%$ per time unit means $P(t) = P_0 (1 + r/100)^t$.
$r > 0$ means growth; $r < 0$ means decay.
(For instance in our city example from class on Tuesday,
where growth rate was $2\%$ per year we had $P(t) = 10(1.02)^t$.)
{\it  On the other hand}, saying $P$ grows or decays at a
{\it  continuous} rate of $k$ means $P$ is written 
in terms of the ``natural exponential'' function
as  $P(t) = P_0 e^{kt}$.  For instance, $k = .10$
might be described as a $10\%$ continuous growth rate.
Similarly $k = -.05$ might be described as a $5\%$ continuous decay
rate.

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\item{1)}  Write the formula for a function for $P(t)$ growing
at a $3\%$ annual rate with $P(0) = 15$.
\item{2)}  Write the formula for a function for $P(t)$ growing
at a $3\%$ continuous annual rate with $P(0) = 15$.
\item{3)}  Are your functions in 1 and 2 the same?  Write your
function in 2 in the form $P(t) = P_0 a^t$ for some $a$ and
compare with 1.

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D) Do problem 38 from Section 1.2 (page 17) of the text.
Explain how you are filling in the table.


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{\it  Assignment} 


One write-up per group of solutions for these problems, 
due Thursday, September 8.

\bye
