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\centerline{MATH 135-- Calculus 1}
\centerline{Rates of Change, Slopes of Tangents}
\centerline{September 16, 2016}
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\noindent
{\it Background}
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We are now ready to move into Chapter 2 of our text and consider average rates of change, and eventually
limits and tangent lines.  \emph{This is really the start of the subject of calculus itself}.  Everything
we have done up until now has been review of precalculus topics to prepare!

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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)]  A ball dropped from a height of $40$ meters and from a state of rest at time $t = 0$ has height 
$s(t) = 40 - 4.9 t^2$ meters at time $t$ seconds.
\begin{enumerate}
\item[(a)]  How far does the ball travel between $t = 1$ and $t = 1.5$?  What is the ball's
average velocity over that interval?  (Note: Your answer should be negative.  That just means
the ball is moving downward.)
\item[(b)]  Complete the following table by following the same steps you did in part (a) for 
each of the given intervals.  Use at least 5 decimal places in all calculations for these.
$$\begin{array}{cccccccccc}
\hline
{\rm interval} &\vline& [1,1.1] &\vline& [1,1.01] &\vline& [1,1.001] &\vline& [1,1.0001]\\ \hline
{\rm ave. vel.}&\vline&         &\vline&          &\vline&           &\vline&           \\ \hline 
\end{array}$$
\item[(c)]  Using this information, estimate the velocity of the ball \emph{right at the instant} $t = 1$.
This is called the \emph{instantaneous velocity} of the ball at that time.
\item[(d)]  Now repeat part (b) for these intervals \emph{ending at} $t = 1$:
$$\begin{array}{cccccccccc}
\hline
{\rm interval} &\vline& [0.9,1] &\vline& [0.99,1] &\vline& [0.999,1] &\vline& [0.9999,1]\\ \hline
{\rm ave. vel.}&\vline&         &\vline&          &\vline&           &\vline&           \\ \hline 
\end{array}$$
Are these results consistent with what you did before in part (b)?  (They should be!)
\end{enumerate}
\item[2)]  Do problem 26 on page 62 in our text.
\item[3)]  Do problem 27, parts (b), (c), (d) on page 62 in our text.  The answer for part (a) would be
``percent infected per day.''
\end{enumerate}
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