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\title{ESP Workshop, Worksheet \#12}
\date{\today}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#12}

{\bf \large Thursday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
\item Hey!  You've got an exam coming up!  Let's practice stuff we know with some of Durbin's old exam problems!
\begin{enumerate}
\item Find the radius of convergence and the interval of convergence for
\begin{displaymath}
\sum _{n=1} ^\infty \frac{(-1)^n (x-2)^n}{n3^n}
\end{displaymath}

\item Find the Taylor series centered at $c=\pi /2$ for $f(x) = \sin
  x$. Express your answer using $\sum$ notation.

\item Using polar coordinates, draw the graphs of $r = 2$ and $r = 2 +
  \sin \theta$. 

\item Let $\mathbf{u} = \langle -5, 12 \rangle$ and $\mathbf{v}
  =\langle 3,4 \rangle$. Find: (i) the angle between $\mathbf{u}$ and
  $\mathbf{v}$, (ii) the projection of $\mathbf{u}$ onto $\mathbf{v}$,
  and (iii) the vector component of $\mathbf{u}$ orthogonal to
  $\mathbf{v}$.
\end{enumerate}

\item We just learned about cylindrical and spherical coordinates in
  three-space. As a warm-up, let's compare some graphs in rectangular
  and polar coordinates in two-dimensional space. Graph each equation
  on its own set of axes.
\begin{enumerate}
\item In rectangular coordinates $(x,y):\ x = 5;\ x^2 + 4y^2 = 1$
\item In polar coordinates $(r,\theta):\ r = 5;\ \theta = \pi/4;\ r = 2 + \sin (4\theta)$
\item Recall that we have the following relationships between polar
  $(r,\theta)$ and rectangular coordinates $(x,y)$:
\begin{displaymath}
r^2 = x^2 + y^2,\ \tan \theta = \frac{y}{x}
\end{displaymath}
\begin{displaymath}
x = r \cos \theta,\ y = r \sin \theta
\end{displaymath}
Can you write the equation $r = 2+\sin (4\theta)$ in rectangular
coordinates? (Hint: it ain't pretty.)
\end{enumerate}

\item Let's practice using the three coordinate systems we know for
  three-dimensional space. For each of the following, describe the
  surface and draw it (if you can).
\begin{enumerate}
\item In rectangular coordinates $(x,y,z):\ x = 5;\ 4x^2 + 4y^2 + z^2
  = 1$
\item In cylindrical coordinates $(r,\theta, z):\ r = 2;\ r = 2 + \sin
  \theta;\ \theta = \pi/2;\ 4r^2 + z^2 = 1$
\item In spherical coordinates $(\rho, \theta, \phi):\ \rho = 2;\ \rho
  = 2 + \sin (4\theta);\ \rho = 2 + \sin (4\phi) $
\item Can you describe a golf ball using spherical coordinates?
\item (A Geography/Zoology question!) A bear is sitting at some point
  $N$ on the Earth. He gets up and walks one mile south, then one mile
  east, and finally one mile north. If the bears ends up at $N$, the
  same point where he started, then what color is the bear?
\end{enumerate}

\item It's solid-sketching time! Also, describe what each solid could
  be used for.
\begin{enumerate}
\item In rectangular coordinates $(x,y,z): 1\leq |x| \leq 2;\ 1 \leq
  |y| \leq 2; 1 \leq |z| \leq 2$
\item In cylindrical coordinates $(r,\theta, z): 1 \leq r \leq 2;\ 0 \leq \theta \leq \pi;\ 0\leq z \leq r-1$
\item In spherical coordinates $(\rho, \theta, \phi): 1 \leq \rho \leq
  2;\ 0 \leq \theta \leq \pi;\ \pi /2 \leq \phi \leq \pi(r-1)/6 + \pi/2 $
\item How are (b) and (c) similar? How are they different?
\end{enumerate}

\begin{figure} \label{gymnastics}
\begin{center}
\includegraphics{worksheet12}
\caption{One the left is a cross-section of a balance beam, and on the
  right is one of the chalk holder.}
\end{center}
\end{figure}

\item Kelly, one of the ESP calculus leaders from last year, is a
  gymnast. Use inequalities in the appropriate coordinate system to
  describe...
\begin{enumerate}
\item ...a tumbling mat (i.e. a very short box).
\item ...the uneven bars.
\item ...the balance beam (assume that the cross-section looks like
  that in Figure 1.
\item ...the rings.
\item ...the chalk-holder (looks like a steel drum---see the
  cross-section in Figure 1).
\end{enumerate}

\item Each edge of a cubical box has length 1 m.  The box contains
  nine spherical balls---eight orange and one white---all with the
  same radius $r$. The center of the white ball is at the center of
  the cube and it touches the other eight balls, each of which is near
  one of the eight corners of the box.  Each of the orange balls
  touches three sides of the box. Thus, the balls are tightly packed
  in the box. Find $r$.

\end{enumerate}

\end{document}