\def\wsn{19}
\input worksheet.tex

\hbox{\vbox{\hsize=4.2in
\item{1.}  Suppose we want to fit a line to a set of data as shown in
the figure to the right.  The {\it least squares method} is based on
minimizing the sum of the squares of the errors.  If we have a set of
data points $(x_i,y_i)$ for $i=1,2,\ldots,n$ this will be a sum of $n$
positive values.  The error for a data point $(x_i,y_i)$ is the
difference between $y_i$ and the height at $x_i$ of the line we
select.

\medskip
\itemitem{a)}  Suppose we choose a line of the form $y=\alpha x$.
Find an expression for $S(\alpha)$, the sum of the squares of the
errors. 

\medskip
\itemitem{b)}  Find a formula for $\alpha$ which minimizes
$S(\alpha)$.} 
\hskip.3in
\epsfxsize=2in\epsfbox{/home/oehrtman/m210/leastsquares.eps}}

\bigskip
\item{2.}  A certain function $f$ defined for all $x$ has
$$
\eqalign{
f''(0)=&0,\cr
f''(x)>&0\quad\hbox{for }x>0,\cr
\hbox{and }f''(x)<&0\quad\hbox{for }x<0.\cr}
$$
The number of critical points, number of local maxima, number of local
minima, and number of roots of $f$ are all tabulated.  Give all
possible such tabulations. 

\bigskip
\item{3.}  In this problem, you are going to trace the shape of your
hand and approximate the area of the picture that you create.  Your
main tasks are to devise a method for approximating the area and to
show that your approximation is very close to the actual area.

\medskip
\itemitem{a)}  Start with a sheet of blank paper and make an accurate
trace of the outline of your hand.

\medskip
\itemitem{b)}  Devise a method to approximate the area of the region
inside the curve you have traced.  You must explain your method in
detail and why it works.

\medskip
\itemitem{c)}  Find a way for estimating the error for the method you
devised.  
\itemitem{}  Note:  Error is something you would like to make {\it
small}!  Thus an estimate for the error means being able to say the
error is {\bf less than} some value.

\medskip
\itemitem{d)}  Find an approximation off the area of your tracing that
differs from the actual area by less than 1\% of the actual area.

\bigskip
\item{4.}  Let $f(x)=(x^2-4)/(x-1)$

\medskip
\itemitem{a)}  Find the intercepts.

\medskip
\itemitem{b)}  Find all vertical asymptotes.

\medskip
\itemitem{c)}  Differentiate and simplify.

\medskip
\itemitem{d)}  Find all points $c$ where $f'(c)=0$ or $f'(c)$ does not
exist. 

\medskip
\itemitem{e)}  Use the first derivative to find where $f$ is
increasing and decreasing.  

\medskip
\itemitem{f)}  Find the second derivative of $f$ and simplify. 

\medskip
\itemitem{g)}  Find all points $c$ where $f''(c)=0$ or $f''(c)$ does
not exist.   

\medskip
\itemitem{h)}  Use the second derivative to find where $f$ is concave
up and concave down.  

\medskip
\itemitem{i)}  Plot all intercepts, places where $f'(c)=0$, where
$f''(c)=0$ and all asymptotes.  

\medskip
\itemitem{j)}  Graph $f$.





\bye