
{\bf Name:}\rule{2in}{.2mm}
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\Large\bf Problem Set 1
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\bf INSTRUCTIONS: Read  all the problems carefully and show all your work.

\begin{problem} 
Temperature readings T (in degrees Celsius) were
recorded every hour starting at midnight on a day in May in Louisville,
Kentucky, as in the following table. The time x is measured in hours
from midnight. Find the average rate of change of temperature with
respect to time for the following: (a) from noon to 3 PM. (b) from noon
to 2 PM. 
\end{problem}

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%\color{red} Solution:  (a) From noon to 3 PM the temperature changes from 14.3 to 18.2.\\
%So $\triangle$T = T(15)-T(12) = 18.2-14.3 = 3.9\\
$\frac{T}{x}$ = $\frac{3.9}{15-12}$ = 1.3\\
Solution: (b) From noon to 2 PM the average rate of change is\\
$\frac{T}{x}$ = $\frac{T(14)-T(12)}{14-12}$ = $\frac{17.3-14.3}{2}$ = 1.5\\  
\begin{problem}
Plot the data on the graph paper given below and draw a plausible temperature function based on the information
in the table.\\
\end{problem}
\begin{tabular}[.5in]{|r|r|}\hline
Time x  &Temp T\\  
(hours)&($C^{\circ}$)\\ \hline
0&6.5\\ \hline
1&6.1\\ \hline
2&5.6\\ \hline
3&4.9\\ \hline
4&4.2\\ \hline
5&4.0\\ \hline
6&4.0\\ \hline
7&4.8\\ \hline
8&6.1\\ \hline
9&8.3\\ \hline
10&10.0\\ \hline  
11&12.1\\ \hline
12&14.3\\ \hline
13&16.0\\ \hline
14&17.3\\ \hline
15&18.2\\ \hline
16&18.8\\ \hline
17&17.6\\ \hline 
18&16.0\\ \hline
19&14.1\\ \hline
20&11.5\\ \hline
21&10.2\\ \hline
22&9.0\\ \hline
23&7.9\\ \hline
24&7.0\\ \hline
\end{tabular} 
\hspace{2in}\vspace{-2in} \graphpaper[1](0,0)(24,19)
 
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\begin{problem}
Use the table to estimate the instantaneous rate of  change at noon?
\end{problem}
%\color{red} Solution: We plot the given data and use them to sketch a smooth curve that approximates the graph of the temperature function.  We draw the tangent at the point x = 12 and, after measuring the sides of the triangle ABC, we estimate that the slope of the tangent line is\\
%$\frac{|BC|}{|AC|}$ = $\frac{10.3}{5.5}$ = 1.9\\
%Therefore, the instantaneous rate of temperature with respect to time at noon is about 1.9.\\
  
