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\title{ESP Workshop, Worksheet \#13}
\date{\today}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#13}

{\bf \large Tuesday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
\item Here are some practice exam-ish problems... it would be a good
  idea to practice writing clean, logical solutions, complete with
  proper symbols.  Ask Eric if you have any questions about this!
\begin{enumerate}
\item Find the radius of convergence and interval of convergence of the series:
\begin{displaymath}
\sum_{n=1} ^\infty (-1)^n \frac{(x+2)^n}{n 2^n}
\end{displaymath}

\item Find the Taylor series for $f(x)$ centered at the given value of $a$.
\begin{displaymath}
f(x) = \ln x,\ a=2
\end{displaymath}

\item Use the binomial series to expand the function as a power
  series. State the radius of convergence.
\begin{displaymath}
\sqrt[4]{1-8x}
\end{displaymath}

\item Show that the equation represents a sphere, and find its center
  and radius.
\begin{displaymath}
x^2 + y^2 +z^2 = x+y+z
\end{displaymath}

\item Let $\mathbf{u} = \langle 2,3 \rangle$ and $\mathbf{v} =
  \langle 0,5 \rangle$. Compute $\mathbf{u} + 2\mathbf{v},
  \mathbf{u}\cdot \mathbf{v},$ and
  $\textrm{proj}_{\mathbf{v}}\mathbf{u}$ and draw pictures for the
  first and last ones. Why doesn't it make much sense to draw a
  picture about the dot product?

\item Let $\mathbf{a} = \langle 1,2,3 \rangle$ and $\mathbf{b} =
  \langle -1, 1, -1 \rangle$. Find the angle between $\mathbf{a}$ and
  $\mathbf{b}$ by computing $\mathbf{a}\cdot \mathbf{b}$ and also by
  computing $\mathbf{a} \times \mathbf{b}$.

\item Find an equation for the plane that passes through the point
  $(6,0,-2)$ and contains the line $x=4-2t,\ y=3+5t,\ z=7+4t$.

\item Plot the point whose cylindrical coordinates are $(4, -\pi/3,
  5)$. Then find the rectangular coordinates of the point.
\end{enumerate}

\item (Fibonacci numbers!) Show that the Maclaurin series of the function
\begin{displaymath}
f(x) = \frac{x}{1-x-x^2}\quad \textrm{is}\quad \sum_{n=1} ^\infty f_n x^n
\end{displaymath}
where $f_n$ is the $n$th Fibonacci number, i.e. $f_1 = 1,\ f_2 = 1,$
and $f_n = f_{n-1} + f_{n-2}$ for $n\geq 3$. [Hint: Write $x/(1-x-x^2)
= c_0 +c_1 x +c_2 x^2 +\cdots$ and multiply both sides of this
equation by $1-x-x^2$.]

Now obtain the Maclaurin series for $f(x)$ in a different way by
writing $f(x)$ as a sum of partial fractions. Use this and your
previous answer to find an explicit formula for $f_n$.

\newpage

\item Sketch the following surfaces:
\begin{eqnarray*}
x^2 + 9z^2 &=& 36 \\
y^2 + 4z^2 &=& 1 + x^2 \\
y^2 &=& 1 + 4x^2 + 9z^2 \\
x + y^2 &=& z^2 \\
x^2 &=& y^2 + z^2
\end{eqnarray*}
\end{enumerate}

\end{document}