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\title{Math 1b. Arc Length 
and the Average Value of a Function}

\date{Spring 2006}

\begin{document}

\maketitle




\begin{enumerate}

\item
Find the length of the curve given by
$$
y = \frac{2}{3} (x^2 - 1)^{3/2}
$$
on the interval $[1, 3]$.

\vfill

\item
The arc length function for a function $f(t)$ beginning at the point $x = a$ is defined by
$$
s(x) = \int_a^x \sqrt{1 + [f'(t)]^2} \, dt.
$$
Calculate $s(x)$ for $y = x^{3/2}$, where $a = 1$.

\vfill

\clearpage

\item
How would you find the average temperature in Boston over a 24-hour period?

\item
How would you calculate the average temperature in Boston given the following data?

\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{| r | l |} \hline
Time & Temperature \\ \hline
2 PM & 68 $^\circ$F \\
6 PM & 68 \\
10 PM & 58 \\
2 AM & 55 \\
6 AM & 54 \\
10 AM & 65 \\
2 PM & 74 \\ \hline
\end{tabular}
\end{center}

\vfill

\item
How would you calculate the average temperature in Boston given that the temperature is given by
$$
f(t) = \frac{1}{9} t^2 - \frac{29}{12} t + 68,
$$
where $t$ is given in hours since 2 PM and $f(t)$ is given in degrees Fahrenheit?

\vfill


\item
The graph of some function $f$ is given below.  List the following values from {\em smallest\/} to {\em largest}.   


\vspace{.2in}
\centerline {
\includegraphics[width=3in]{Fig_11_1}
}
\vspace{.2in}


\begin{enumerate}

\item
$f'(1)$

\item
The average value of $f(x)$ for $0 \leq x \leq a$.

\item
The average value of the rate of change in $f(x)$ for $0 \leq x \leq a$.

\item
$\displaystyle \int_0^a f(x) dx$.

\end{enumerate}



\end{enumerate} 



 \end{document}