\def\wsn{18}
\input worksheet.tex

\noindent{\bf Chain Rule}

\bigskip
\item{1.}  Suppose that $f$ and $g$ are two functions such that
$f(g(x))=x$.  Show that
$$
g'(x)={1\over f'(g(x))}{\atop.}
$$

\bigskip
\item{2.}  Let $y=\tan^2(\arccos(x))$.

\medskip
\itemitem{a)}  Find $dy/dx$ using the chain rule.

\medskip
\itemitem{b)}  Find another expression for $y$ which does not mention
trig functions.  (Hint:  Draw a triangle.)

\medskip
\itemitem{c)}  Find $dy/dx$ from your expression in part b).  How does
it compare with your answer in part a)?

\bigskip
\noindent{\bf Implicit Differentiation}

\bigskip
\item{3.}  A differentiable function $y(x)$ satisfies $x^2\cos y+\sin
y=x$ and $y(1)=0$.  What is $y'(1)$?

\bigskip
\item{4.}  Prove that the tangent line to a circle at any point is
perpendicular to the radius to that point. 


\bigskip
\noindent{\bf Related Rates}

\bigskip
\item{5.}  Given that a spherical raindrop evaporates at a rate
proportional to its surface area, how fast does the radius shrink?

\bigskip
\item{6.}  A boat is pulled in to a dock by means of a rope with one
end attached to the bow of the boat, the other end passing through a
ring attached to the dock at a point 4 feet higher than the bow of the
boat.  If the rope is pulled in at a rate of 2 feet per second, how
fast is the boat approaching the dock when 

\medskip
\itemitem{a)} 10 feet of the rope are out?

\medskip
\itemitem{b)} 5 feet of rope are out?

\bigskip
\hbox{\vbox{\hsize=4in\baselineskip=16pt
\noindent{\bf Max/Min Problems}

\bigskip
\item{7.}  Suppose you are given a circle of radius $r$ and a tangent
line $L$ to the circle through a point $P$ on the circle.  From a
variable point $R$ on the circle, a perpendicular $PQ$ is drawn to
$L$ with $Q$ on $L$.  Determine the maximum of the area of triangle
$PQR$.\bigskip}\hskip.3in
\epsfxsize=2in
\epsfbox{/home/oehrtman/m210/circle18.eps}}

\bigskip
\item{8.}  Joel has fallen off of his ``Little Mermaid'' floaty 200
feet from shore in Lake Travis.  He cannot swim.  Cheryl is at a point
200 feet down the shore from the point closest to Joel.  She can run
18 ft/s and can swim at a rate of 5 ft/s.  

\medskip
\itemitem{a)}To what point on the shore should she run before diving
into the lake {\bf if} she wants to reach Joel as quick as possible?

\medskip
\centerline{\epsfxsize=2.8in\epsfbox{/home/oehrtman/m210/drowning.eps}}

\medskip
\itemitem{b)}  Once Joel falls into the water, he can manage to thrash
about for one minute.  It takes Cheryl 10 seconds to notice Joel is in
trouble.  Can she reach him in time?


\bye