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\centerline{Mathematics 241 -- Multivariable Calculus}
\centerline{Discussion 2 -- Applications of Parametric Curves}
\centerline{September 22, 2010}
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\noindent
{\it Background}
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Today we will look at several more examples with parametric curves. 
At the end of the period we will reconvene and discuss your solutions.  
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\noindent
I.  A slide has the shape of a helix--the curve
$$\alpha(s) = (a\cos(bs),a\sin(bs),cs)$$
for $s\in {\bf R}$ and some constants $a,b,c$.  The radius is
of the slide is $5$ feet and it has height $50$ feet, making 
5 complete turns about the $z$ axis as it descends to the ground.  
To get to the top of the slide, you climb a ladder that goes 
straight up the vertical axis of the slide to a platform at the top.
Suppose that it takes 2 minutes to climb the $50$ feet,
15 seconds to walk to the edge of the platform, then
30 seconds to slide to the bottom.  Assume that 
each portion of the trip is made at a constant speed
(different for each portion).  Construct a compound
(that is, piecewise) parametrization
of the whole motion. 
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\noindent
II.  Anoter way parametric curves often come into play is this:
When we have several related quantities changing with respect
to time, it is often instructive to look both at individual
plots versus time, and to look at what is sometimes called a
{\it phase portrait} that describes the evolution of the system
together.  If we have two quantities $x,y$ changing with respect
to time, then the phase portrait will be the curve $x = x(t), y = y(t)$, 
an example of a {\it parametric curve}, like the lines we have studied
before.  Here is a realistic
example, adapted from p. 65 of our book.  Part of the human immune
system are specialized cells called killer T-cells that attack
other cells that have been infected by viruses.  
The points that are plotted in the graph on the back are from 
a simulation of the response to a viral infection in the 
bloodstream of a human patient.

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To interpret what this is saying, 
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\item{1)}  The coordinates are pairs $(I, K)$ where $I = $
number of infected cells per ml of blood in hundreds and $K = $
number of killer T-cells from the immune system per ml of blood,
also in hundreds.  
\item{2)}  The initial measurement was $(I, K) = (5,2)$ made at
time $t = 0$.
\item{3)}  The subsequent points were measurements at {\it equal
time intervals} of one hour.
\item{4)}  As $t$ increases, the later states of the system
are given by points that appear to move {\it counterclockwise} in 
the $(I,K)$-plane.

\vglue 3.2truein
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\noindent
{\it Questions}
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\item{A)}  From the given $(I,K)$ data points, sketch plots
of $K$ versus $t$ and $I$ versus $t$.  Try to
``connect the dots'' to get smooth curves here.  
\item{B)}  When do $I$ and $K$ reach their absolute maximum and minimum values?
When do they have local maxima and minima?
\item{C)}  Where in the phase portrait (the $(I,K)$ data plot) are
the infected cell and T-cell counts {\it changing the fastest}?  Explain
the reasoning behind your answer.
\item{D)}  What happens in the long run?  Does the immune response
kill off all of the infected cells?  Does the infection "explode"
and grow unchecked?  Or is there a {\it chronic} infection (one that 
stays at a nonzero level over a long time period)?
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\bye
