\def\wsn{10}
\input worksheet.tex

\centerline{\bf Mike's Bizarre Function Zoo}

\bigskip
\itemitem{1.\hskip12pt a)}  Graph $f(x)=\root3\of x$ and its tangent
line at $x=0$.  

\medskip
\itemitem{b)}  Find the slope of a secant line to $f$ intersecting the
graph when $x=0$ and $x=h$.

\medskip
\itemitem{c)}  Find the limit of these slopes as $h\to0$.

\medskip
\itemitem{d)}  Find $\displaystyle\lim_{x\to0}f'(x)$.

\medskip
\itemitem{e)}  What is the equation of the tangent line to the graph
of $f$ at $x=0$?

\bigskip
\item{2.}  Let $\displaystyle{s(x)=\cases{e^{-1/x^2}&$x\neq0$;\cr
c&$x=0$,\cr}}$\hskip.2in where $c$ is a constant.

\medskip
\itemitem{a)}  Find the value of $c$ for which $s$ is continuous.

\medskip
\itemitem{b)}  Use your calculator to compute the limit $s'(0)$. 

\bigskip
\item{3.}  Let $\displaystyle{g(x)=\cases{x\sin(1/x)&$x\neq0$;\cr 
a&$x=0$,\cr}}\qquad$ and $\qquad\displaystyle
{h(x)=\cases{x^2\sin(1/x)&$x\neq0$;\cr b&$x=0$.\cr}}$

\medskip
\itemitem{a)}  Find the values of $a$ and $b$ for which $g$ and
$h$ are continuous functions.

\medskip
\itemitem{b)}  Find the limits $g'(0)$ and $h'(0)$.

\medskip
\itemitem{c)}  Sketch graphs of $g$ and $h$.  Explain geometrically
your results to part b).

\bigskip
\item{4.}  Let $\displaystyle{r(x)=\cases{1&$x$ is rational;\cr 
0&$x$ is irrational.\cr}}$  

\medskip
\itemitem{a)} Sketch a graph of $r$.  Is it continuous?  Why or why
not?

\medskip
\itemitem{b)}  Sketch several secant lines through $x=0$ and $x=h$
where $h$ is very small.  Find the limit $r'(0)$.

\bigskip
\item{5.}  Let $\displaystyle{v(x)=\cases{x^2&$x$ is rational;\cr 
0&$x$ is irrational.\cr}}$  

\medskip
\itemitem{a)} Sketch a graph of $v$ and show that it is continuous at
$x=0$. 

\medskip

\itemitem{b)}  Show that $v$ is diffferentiable at $x=0$.

\bigskip
\item{6.}  Let $\displaystyle{w(x)=\cases{1/q&$x\neq0$ is rational and
$x=p/q$ is in simplest form;\cr 
0&$x=0$;\cr
0&$x$ is irrational.\cr}}$

\medskip
\itemitem{a)}  Fill in the following chart:

\medskip
\hbox{\hskip.5in
\vrule\vbox{\hsize=6.1in\leftskip=.2in\hrule
\item{$x$}\vrule height15pt depth5pt
\quad0\quad3\quad3.1\quad3.14\quad3.141\quad3.1415\quad$\pi$\quad
1\quad1.4\quad1.41\quad1.414\quad$\sqrt2$\quad
2\quad2.7\quad2.71\quad2.718\quad$e$
\hrule
\item{$w(x)$}\vrule height15pt depth5pt
\hrule}\vrule}

\medskip
\itemitem{b)}  Sketch a graph of $w$.  Where is it continuous?

\medskip
\itemitem{c)}  Where is $w$ differentiable?


\bye