\def\wsn{9}
\input worksheet.tex

\item{1.}  For each of the following descriptions of a function $f$,
sketch the graph of such a function.  If no such function exists,
explain why.

\medskip
\itemitem{a)}  $f'(a)$ exists and $f$ is continuous at $a$.

\medskip
\itemitem{b)}  $f'(a)$ exists but $f$ is not continuous at $a$.

\medskip
\itemitem{c)}  $f'(a)$ does not exist but $f$ is continuous at $a$.

\medskip
\itemitem{d)}  $f'(a)$ does not exist and $f$ is not continuous at $a$.

\medskip
\item{}  If any of your examples have a cusp at $a$, find an example
for that description which does not have a cusp.  

\medskip
\item{}  If, on the other hand, none of your examples have a cusp at
$a$, indicate which of these four descriptions would be appropriate
for such a function.

\bigskip
\item{2.}  For each graph of a function $f$ below, sketch a graph of
its derivative $f'$.

\medskip
\vbox{
\hbox{
\vbox{\hsize=.5truein a)\vskip1truein}
 \epsfxsize=1.2truein\epsfbox{/home/oehrtman/m210/graph02.a.eps}
\hskip.5in\vbox{\hsize=.5truein b)\vskip1truein}
 \epsfxsize=1.2truein\epsfbox{/home/oehrtman/m210/graph02.b.eps}
\hskip.5in\vbox{\hsize=.5truein c)\vskip1truein}
 \epsfxsize=1.2truein\epsfbox{/home/oehrtman/m210/graph02.c.eps}}
\vskip.2truein
\hbox{
\vbox{\hsize=.5truein d)\vskip1truein}
 \epsfxsize=1.2truein\epsfbox{/home/oehrtman/m210/graph02.d.eps}
\hskip.5in\vbox{\hsize=.5truein e)\vskip1truein}
 \epsfxsize=1.2truein\epsfbox{/home/oehrtman/m210/graph02.e.eps}
\hskip.5in\vbox{\hsize=.5truein f)\vskip1truein}
 \epsfxsize=1.2truein\epsfbox{/home/oehrtman/m210/graph02.f.eps}}
\vskip.2truein
\hbox{
\vbox{\hsize=.5truein g)\vskip1truein}
 \epsfxsize=1.2truein\epsfbox{/home/oehrtman/m210/graph02.g.eps}
\hskip.5in\vbox{\hsize=.5truein h)\vskip1truein}
 \epsfxsize=1.2truein\epsfbox{/home/oehrtman/m210/graph02.h.eps}
\hskip.5in\vbox{\hsize=.5truein i)\vskip1truein}
 \epsfxsize=1.2truein\epsfbox{/home/oehrtman/m210/graph02.i.eps}}}

\bigskip
\item{3.}  Given below is a {\it one-parameter family of functions}.
That is, it is a collection of many functions $f_t$ - one for each
value of the parameter $t$. 
$$
f_t(x)=\cases{t\sin x,&$x\leq{\pi\over2}$;\cr\cr
(1-t)x+t^2,&$x>{\pi\over2}$.\cr} 
$$

\itemitem{a)}  Determine which values of the parameter $t$ give a
continuous function.  Graph $f_t$ for each of these values of $t$.

\medskip
\itemitem{b)}  Determine which values of the parameter $t$ give a
differentiable function.  Looking at your graphs from part a), give a
geometric explanation of your results.  

\bigskip
\item{4.}  The equation of the tangent line to the graph of a function
$f$ at $x=4$ is $y=3x-17$.

\medskip
\itemitem{a)}  What is $f(4)$?  What is $f'(4)$?

\medskip
\itemitem{b)}  Given that $f(x)=ax^3+b$, find the constants $a$ and
$b$.

\bigskip
\item{5.}  Prove starting from the the definition of the derivative
and stating any assumptions made about properties of limits (also draw
a picture to illustrate):

\medskip
\itemitem{a)} if $g(x)=f(x)+c$, then $g'(x)=f'(x)$;

\medskip
\itemitem{b)} if $g(x)=cf(x)$, then $g'(x)=cf'(x)$;

\medskip
\itemitem{c)} if $g(x)=f(x)+h(x)$, then $g'(x)=f'(x)+h'(x)$.

\bigskip
\itemitem{6.\hskip12pt a)}  Find all points on the graph of $y=x^2$
whose tangent line passes through the point $(5,0)$.

\medskip
\itemitem{b)}  Show that no line tangent to the graph of
$\displaystyle f(x)=x+{1\over x}$ passes through the origin.

\bigskip
\item{7.}  Find the equation to the line tangent to the graph of
$y=x^{1/3}$ at the origin.



\bye