\def\wsn{7}
\input worksheet.tex

\item{1.}  Compute the following limts:
$$
\pt a \lim_{x\to0}{\tan^23x\over5x}\hskip.5in
\pt b \lim_{x\to1}{\sin(x^2-1)\over x-1}\hskip.5in
\pt c \lim_{x\to-1}{\sin(x^2-1)\over x-1}\hskip.5in
\pt d \lim_{x\to0}{\sin x^2\over\sin^22x}
$$

\item{2.}  Find examples of functions $f$ and $g$ satisfying the
following: 

\smallskip
\itemitem{a)} $\displaystyle\lim_{x\to c}g(x)=0$ but $\displaystyle
{f(x)\over g(x)}$ does not have a vertical asymptote at $x=c$.

\smallskip
\itemitem{b)}   $\displaystyle\lim_{x\to c}f(x)=0$ and $\displaystyle
{f(x)\over g(x)}$ does have a vertical asymptote at $x=c$.

\smallskip
\itemitem{c)} $\displaystyle\lim_{x\to c}g(x)\neq0$ and $\displaystyle
{f(x)\over g(x)}$ does have a vertical asymptote at $x=c$.

\smallskip
\itemitem{d)} $\displaystyle\lim_{x\to c}f(x)$ exists and is not zero
and $\displaystyle\lim_{x\to c}g(x)=0$.  Does the graph of
$\displaystyle {f(x)\over g(x)}$ have a vertical asymptote at $x=c$?

\medskip
\itemitem{3.\hskip12pt a)}  Find the slope of the secant line
intersecting the graph of $f(x)=\sin x$ at $x$ and $x+h$.

\smallskip
\itemitem{b)}  Simplify your expression using the identity 
$$
\sin(A+B)=\sin A\cos B+\sin B\cos A.
$$ 

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

\medskip
\item{4.}  Consider a fixed point $(a,b)$ on the unit circle.  Now let
$(x,y)$ represent a varying point on the unit circle.

\smallskip
\itemitem{a)}  Give algebraic conditions for $(a,b)$ and $(x,y)$ to
lie on the unit circle.

\smallskip
\itemitem{b)}  Find the slope of the secant line between these two points.

\smallskip
\itemitem{c)}  What happens to your expression for the slope of the
secant line as $(x,y)\to(a,b)$?

\smallskip
\itemitem{d)}  Find the slope of the tangent line to the circle at
$(a,b)$.  

\itemitem{} Hint:  Multiply by a clever form of one which makes the
numerator a difference of squares.  Then use your equations from part
a) to rid the numerator of all $b$'s and $y$'s.  Then$\ldots$


\medskip
\item{5.}  Let $\displaystyle f(x)=\cases{1&$x\geq2$;\cr x&$x<2$.\cr}$

\smallskip
\itemitem{a)}  Find a formula giving the slope of the secant line
intersecting $f(x)$ when $x=2$ and $x=2+h$.

\smallskip
\itemitem{} Note: You will have different answers for $h<0$ and
$h>0$.  Why?

\smallskip
\itemitem{b)}  Find the limit of these slopes as $h\to0^-$ and $h\to0^+$.

\medskip
\item{6.}  When a model rocket is launched, the propellant burns for a
few seconds, accelerating the rocket upward.  After burnout, the
rocket coasts upward for awhile and then begins to fall.  A small
explosive charge pops out a parachute shortly after the rocket starts
down.  The parachute slows the rocket to keep it from breaking when it
lands.

\hbox{\vbox{\hsize=4.1in
\item{}  The figure to the right shows velocity data from the flight
of a model rocket.  Use the data to answer the questions below.

\smallskip
\itemitem{a)}  How fast was the rocket climbing when the engine stopped?

\smallskip
\itemitem{b)}  For how many seconds did the engine burn?

\smallskip
\itemitem{c)}  When did the rocket reach its highest point?  What was
its velocity there? 

\smallskip
\itemitem{d)}  When did the parachute pop out?  How fast was the
rocket falling then?

\smallskip
\itemitem{e)}  How long did the rocket fall before the parachute opened?

\smallskip
\itemitem{f)}  Very carefully sketch a graph of the height of the
rocket (in feet) versus time (in seconds).}
\hskip.3in
\epsfxsize=2.3in\epsfbox{/home/oehrtman/m210/rocket.ps}}



\bye