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\title{ESP Workshop, Worksheet \#}
\date{\today}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#2}

{\bf \large Tuesday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
\item Let $f:\mathbb{R}_{\geq 0}\rightarrow \mathbb{R}$ be a function
  with domain the non-negative real numbers defined by an integral in
  the following way: for each real number $c$, define
\begin{displaymath}
f(c) = \int _{-c} ^c \sin x\ dx
\end{displaymath}
\begin{enumerate}
\item What famous function is $f$ equal to? Draw a picture to justify
  your answer.
\item What is $f'(c)$? What about $\lim _{c\rightarrow \infty} f(c)$?
\item What does $\int _0 ^\infty \sin x\ dx$ mean? Is it convergent or divergent?
\item Professor Durbin claimed in class on Friday that $\lim
  _{c\rightarrow \infty} \int _{-c} ^c \sin x\ dx$ exists but $\int
  _{-\infty} ^\infty \sin x\ dx$ is divergent. Do you agree?\footnote{Remember,
  he also claimed that ``1/3 of all integrals converge,'' so it may be
  okay to disagree with him from time to time...} What is going on?

\end{enumerate}

\item Jessie, the brainiac from Saved by the Bell, says, ``I can't
  figure out the next problem and I have a test tomorrow and just in
  case I make the worst possible career choice ever, I need to pass!''
  Let's help Jessie by setting $f(x) = (\sin x)/x$, $g(x) = 1/x$, and
  $h(x) = f(g(x))$.

  \begin{enumerate}
  \item Sketch $y=h(x)$, that is, sketch $y = x \sin (1/x)$. Jessie
    jokes, ``now that's what I call a bad hair day!''
    
  \item Are $f$ and $g$ continuous everywhere? Jessie reminds us that
    continuous basically means ``not broken''.
  \item Is $h$ continuous everywhere? Does this contradict your answer
    to part (b)? Jessie wonders what can be said in general about the
    composition of continuous functions---what do you think?
  \item Sketch $y = h'(x)$ without actually calculating $h'(x)$. Jessie
    remembers, ``Oh yeah! $h'(x)$ is just the slope of the line tangent
    to the graph of $y = h(x)$ at the point $(x, h(x))$!''
    
  \item (Challenge.) Jessie has a favorite function, which she calls
    $J(x)$, but she won't tell you what it is. All she'll tell you is
    that $\lim _{x\rightarrow 0} J(x) = 0$. Why couldn't you use
    L'Hospital's rule to evaluate $$\lim _{x\rightarrow 0}
    \frac{J(x)}{h(x)}?$$
  \end{enumerate}
  
  
\item Let $f(x) = \lfloor x \rfloor$. This is called the ``floor''
  function, and it returns the largest integer less than or equal to
  $x$. Also, let $g:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}$ be
  defined by $g(x) = (f(x))!$.

\begin{enumerate}
  
\item Draw the graph of $y=f(x)$ for $x$ in the range $-3 \leq x \leq
  3$. What is $f(-1.5)$? What about $f(1.9999)$?  And
  $f(1.\overline{9})$?

\item Draw the graph of $y=g(x)$ for $x$ in the range $0\leq x \leq 4$.

Let's define a new function, called $\Gamma(x)$, like this:
\begin{displaymath}
\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}\ dt,\qquad x>0.
\end{displaymath}

\item Use integration by parts to prove that $\Gamma(x+1)=x\Gamma(x)$.

\item Show that $\Gamma(1)=1$.  Conclude that $\Gamma(n)=(n-1)!$ for
  all natural numbers $n$.
  
  The gamma function provides an example of a function (continuous on
  the positive real numbers) which {\bf interpolates} the values of
  $n!$ for natural numbers $n$.
  
\item (Challenge.) Show that $\Gamma(1/2) = \sqrt{\pi}$.\footnote{This is
  probably really hard or impossible at this point. We'll come back to
  it after we have seen some multi-variable calculus...}

\end{enumerate}

\item 
\begin{enumerate}
\item Evaluate $\int _1 ^\infty \frac1x\ dx$, draw the graph of $y =
  1/x$ and indicate (graphically) what this integral is measuring.

\item For what values of $p$ does $\lim _{t\rightarrow \infty} t^{1-p}$ converge? Diverge?

\item Suppose $p < 1$. What is
\begin{displaymath}
\int _1 ^\infty \frac{1}{x^p}\ dx ?
\end{displaymath}

\item Now suppose $p > 1$. What is $\int _1 ^\infty x^{-p}\ dx$? What
  about
\begin{displaymath}
\int _1 ^\infty 2\pi x^{1-p}\ dx
\end{displaymath}
and what does it measure? (Hint: what's Eric's least-favorite hole in
putt-putt?) Draw a picture!

\end{enumerate}
  
\item 

There are some ESP students all standing in a circle.

\begin{enumerate}
\item First suppose that every guy is next to two girls and every girl
  is next to two guys (i.e. the circle alternates). If there are 25
  girls, how many guys must there be?
  
\item (Challenge.) Those same students rearrange themselves in some
  crazy order (not necessarily alternating). Prove that both neighbors
  of at least one student are guys.

\end{enumerate}

\end{enumerate}

\end{document}