\def\wsn{2}
\input worksheet.tex

\hbox{\vbox{\hsize=4.1in
\item{1.}  The figure on the right shows a multiflash photograph of
two balls falling after being released from rest.  The rulers in the
figure are marked in centimeters.  The formula for free fall near the
Earth's surface is $s=4.9t^2$, where $s$ is distance fallen (in meters)
and $t$ is the time elapsed (in seconds).

\medskip
\itemitem{a)}  How long did it take the balls to fall the first 160
cm?  What was their average velocity for this period?

\medskip
\itemitem{b)}  How long was the time between consecutive flashes?

\medskip
\itemitem{c)}  Estimate the velocity of the balls after they had
fallen 50 cm and also after they had fallen 160 cm.

\medskip
\itemitem{d)}  How could the experiment be changed so that the
estimates in part c) would be more accurate?  Is there a limit to the
accuracy you could obtain in this way?

\bigskip
\item{2.}    {\it Rectilinear motion} is motion of an object along a
straight line.  Examples of such motion are objects falling under the
pull of gravity, a dragster racing along a straight line course, or an
airplane flying at level flight along a straight line path.

\item{}
Suppose we have an equation which gives the position of a moving
object along a straight line with respect to some reference point
labeled $0$.  Let
$$
s(t)=16t^2
$$
denote the number of feet an object has fallen under the pull of
gravity after $t$ seconds has passed from the first moment it was
first let go.

\medskip
\itemitem{a)}  Make a chart giving the distance that the object has
fallen after $t$ seconds for values $t=0,{1\over2},1,1{1\over2},$ and
$2$.  Determine the {\it average velocities} for the first four
half-second intervals.  From the chart alone, can you determine the
{\it instantaneous velocity} at any time, say at $t=1$?

\medskip
\itemitem{b)}  Write a formula for the average velocity of the object
starting at an arbitrary time $t$ and lasting for $h$ seconds.  From
this formula, can you determine the instantaneous velocity at any
time?

\medskip
\itemitem{c)}  Define the term {\it instantaneous velocity}.
\bigskip}\hskip.5in
\epsfxsize=1.7truein\epsfbox{/home/oehrtman/m210/fallingballs3.ps}}

\bigskip
\item{3.}  When a chemical reaction was allowed to run for $t$
minutes, it produced the amounts $A(t)$ of substance shown in the
following table

\medskip
\centerline{\epsfxsize=4truein\epsfbox{/home/oehrtman/m210/reaction.eps}}

\medskip
\itemitem{a)}  Find the average rate of the reaction for the interval
from $t=20$ to $t=30$.

\medskip
\itemitem{b)}  Plot the data points from the table, draw a smooth
curve through them, and estimate the instantaneous rate of the
reaction at $t=25$.

\eject
\item{4.}  Use a calculator to fill in the table given and then make a
guess at the given limit.


\bigskip
\hbox{\vbox{\hsize=0.6in
\itemitem{a)}\vskip0.5in} 
\epsfxsize=5truein\epsfbox{/home/oehrtman/m210/limit1.eps}} 

\medskip
\hbox{\vbox{\hsize=0.6in
\itemitem{b)}\vskip0.5in} 
\epsfxsize=5truein\epsfbox{/home/oehrtman/m210/limit2.eps}}

\medskip
\hbox{\vbox{\hsize=0.6in
\itemitem{c)}\vskip0.5in} 
\epsfxsize=5truein\epsfbox{/home/oehrtman/m210/limit3.eps}}

\medskip
\hbox{\vbox{\hsize=0.6in
\itemitem{d)}\vskip0.5in} 
\epsfxsize=5truein\epsfbox{/home/oehrtman/m210/limit4.eps}}

\bigskip
\item{5.}  The number of gallons of water in a tank $t$ minutes after
the tank has started to drain is $Q(t)=200(30-t)^2$. 

\medskip
\itemitem{a)}  What is the average rate at which the water flows out
during the first ten minutes? 
\itemitem{}  during the five minutes from $t=5$ to $t=10$?  
\itemitem{}  during the two minutes from $t=8$ to $t=10$?  
\itemitem{}  during the minute from $t=9$ to $t=10$?  

\medskip
\itemitem{b)}  Estimate how fast the water is running out of the tank
at the end of ten minutes.

\medskip
\itemitem{c)}  Draw a graph of the function $Q$ for $0\leq t\leq20$.
Draw the secant lines for the four time intervals used in part a).
What are their slopes?

\bigskip
\item{6.}  Define {\it slope} for the following cases:

\medskip
\itemitem{a)}  for two points

\medskip
\itemitem{b)}  for a line

\medskip
\itemitem{c)}  for an arbitrary curve in the plane





\bye