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\title{ESP Workshop, Worksheet \#1}
\date{Thursday, August 31, 2006}
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{\bf \Large ESP Workshop, Worksheet \#1}

{\bf \large Thursday, August 31, 2006}
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\vspace{.15in}
\begin{enumerate}
\item The year is 1993. Kelly is totally freaking out because she has
  to understand calculus during Saved by the Bell: The College Years!
  Let's help Kelly and A.C. Slater remember some of the stuff they
  learned at Bayside High!
\begin{enumerate}
\item Sketch the graphs of $y=\sin x$ and $y=\csc x$ on the same set
  of axes, and label a few points on each graph. Repeat with $y = \pi
  \cot x$ and $y = \pi \tan x$. Kelly says, ``They look like waves!''.
  Slater says, ``They look like my muscles!'' Who is right?
  
\item Give an example of a function $f:\mathbb{R} \rightarrow
  \mathbb{R}$ such that $$\lim _{x\rightarrow \infty} f(x)$$
  does not
  exist and is not $\pm \infty$. Kelly says, ``Infinity is big!'' and
  Slater says, ``Maybe, but nothing is bigger than my muscles!''
  
\item What is $\lim _{x\rightarrow \infty} \frac{\sin x}{x}$? Can you
  use L'Hospital's rule here? Kelly asks, ``Is that French for 'the
  hospital'?'' Slater replies, ``Yeah, that's where the guy that I
  L'Pinned yesterday had to L'Go.''
  
\item Suppose that $\alpha > 0$. Show that $\lim _{x\rightarrow
    \infty} x^\alpha$ is unbounded. Slater says, ``My opponents' pain
  is unbounded!''
  
\item Again, let $\alpha > 0$. What is $$\lim _{x\rightarrow \infty}
  \frac{x^\alpha \sin x}{e^x}?$$
  Kelly and Slater say, ``Math is
  easy!'' Do you agree?

\end{enumerate}

\item Zack is getting jealous, so we have to teach him some calculus,
  too. Let $f(x) = 2x - 1$ and $g(x) = 1/x$.
  \begin{enumerate}
  \item Sketch the graphs of $y = f(x), y = g(x),$ and $y=f(x)g(x)$ near
    $x=1$. Zack says, ``The graph of $g$ looks like my gorgeous hair!''
    
  \item Given your sketch from part (a), what would you guess the value of
    $$\lim _{x\rightarrow 1} (fg)(x)$$ is? Can you justify your guess?
  \item Can you use L'Hospital's rule to evaluate $\lim _{x\rightarrow
      1} \frac{2x-1}{x}$?
  \end{enumerate}
  
\item Jessie, the brainiac, 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 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 $$\lim _{x\rightarrow 0} \frac{h(x)}{J(x)}?$$
  \end{enumerate}
  
\item Screech loves math, and he has figured out a way to send a
  secret calculus message from the past! Help me decode it! First, let
  $f(x) = \frac12 e^x$.
  \begin{enumerate}
  \item Sketch $y=f(x), y= -f(-x),$ and $y= f(x) - f(-x)$ on the same
    axes. Screech says, ``Sometimes, I Need Help!''
  \item What is $d/dx (f(x) - f(-x))$?
  \item Sketch $y=f(x), y=f(-x),$, and $y = f(x) + f(-x)$ on the same
    axes. Screech says, ``College is Obviously So Hard!''
  \item What is $d/dx (f(x) + f(-x))$?
  \item What do you notice about your answers for parts (b) and (d)? Does
    this phenomenon remind you of another famous pair of functions?
  \item What is Screech trying to tell you??
  \end{enumerate}
  
\item   
  Zack and the gang are throwing a huge party tomorrow, and they need
  your help. The celebration is the most important party they have
  ever hosted. They have 1000 2-liter bottles of soda that they
  are planning to open for the celebration, but they know that one of
  them is poisoned, and they want you to find out which one it is.  
  The poison, although harmless, exhibits no symptoms until ten to
  twenty hours after ingestion, at which point whoever drank it
  becomes ``uncool'', even if that person drank just a little bit of
  the poisoned soda.  
  You have lots of friends willing to try the soda, because Zack has
  promised that all your friends---or at least the ones who are
  ``still cool''---can be extras on the set of Saved by the Bell if
  you find the poisoned soda.  
  What is the smallest number of friends you must have to drink from
  the bottles to be absolutely sure to find the poisoned bottle within
  24 hours?

\end{enumerate}

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