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\centerline{\bigrm Henry and the Fan}
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\noindent
{\bf Purpose:} This project will give you an introduction to maximizing
and minimizing functions.  Although you will not learn a general
technique, you will use what you know about graphing functions and
solving equations to find the maximum of a family of functions.

\noindent
{\bf Procedure:}  You are to follow the steps below.  Do the steps in
order because what you learn in one step will often be needed in the
next step.  After completing the outline below you should write your
results and calculations {\bf neatly}.  Be sure to include graphs
in what you turn in. Your grade will be determined
in part by your presentation.

\noindent
{\bf Project Outline:}

\item{1.}Henry has a fan connected to a power supply as indicated in the
simple circuit shown in Figure 1.  Note that the fan acts as a simple
resistor.  Henry wishes to add a variable resistor to the circuit as
show in in Figure 2.  His problem is that the variable resistor (which
is located in the wall) may
dissipate energy so fast that a fire may start.
In fact, if the power
dissipated through the variable resistor is at least $1\over 3$ the
power used by the fan in Figure 1, then there is danger of starting a
fire.  Any less than $1\over 3$ is fine.
Of course, if the variable resistor is set at a high resistance then not
much current flows and little power is dissipated.  If the resistance
is set very low, it is essentially as if the variable resistor is not
there and the power dissipated is essentially zero.  You want to be sure
that at any intermediate setting the power dissipated is no more than
$1\over 3$ the power consumed by the fan in Figure 1.  Is the circuit in
Figure~2 safe?
 \itemitem{a.}Use Kirchhoff's law and the formula for power to relate the
              constants $V_0$, $R_0$, $I_0$, and $P_0$ which represent
              voltage, resistance, current, and power respectively in
              Figure 1.
 \itemitem{b.}Use Kirchhoff's law and the formula for power on Figure~2
              to relate
              the constants $V_0$ and $R_0$ and the variables $R$,
              $I$, and $P$ where $R$ is resistance of the variable
              resistor, $I$ is the current in the circuit, and $P$ is
              the power dissipated by the variable resistor.
 \itemitem{c.}Combine the formulas of part a to write $P$ as a
              function of $V_0$, $R$, and $R_0$.
 \itemitem{d.}Let $x={R\over R_0}$ and rewrite the formula in part c so
              $P$ is written as $P_0f(x)$ where $f$ is some function of
              $x$. (The formula for $f$ should involve only $x$, not
              $R$ or $R_0$.)
 \itemitem{e.}Use Mathematica to plot the function $y=f(x)$ for a wide
              range of values for $x$. Looking at
              the graph, guess approximately the value of $x$ which makes
              $y$ as large as possible. Then make another plot using a
              smaller domain for the values of $x$. Continue doing this
              until you feel you can determine the value of $x$ which
              makes $f(x)$ as large as possible.
 \itemitem{f.}Now prove that your answer in part e is correct. One way
              to do this is to
              assume there is some $x$ which makes $f(x)$ larger
              than
              the value you obtained in part e and arrive at a
              contradiction. (Answer the original question concerning the
              circuits.)
\item{2.}Now you are to use the techniques you learned in part 1 to
         maximize (or minimize) two other functions.
         Do parts e and f above using the functions $f(x)={x+2\over (x+3)^2}$
         and $f(x)={x+1\over (2x+1)^2}$.  Note that on one the
         asymptote goes up instead of down, so try to find the smallest
         value for $y$ instead of the largest for that functions.
\item{3.}In this step you are to generalize the techniques
         you learned in parts~1 and~2.
         Consider the function $f(x)={(ax+b)\over (cx+d)^2}$ where
         $ad\not=bc$, $c\not=0$, and $a\not=0$.
 \itemitem{a.}Find conditions on the numbers $a$, $b$, $c$, and $d$
              which determine if the vertical asymptote goes up or goes
              down.
 \itemitem{b.}Find a formula for $x$ in terms of $a,b,c,d$ which makes
              $f(x)$ as large as possible in the case of the asymptote
              going down and as small as possible in the case of the
              asymptote going up.  (Try to use what you learned in part
              2.  It may be useful to use Mathematica for solving
              certain equations.)

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