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Recitation 5\hfill \course\\
10 September 1998 \hfill \semester \\
\medskip

Below is a selection of problems related to section 1.5.
These problems will not be collected or graded. However, you should
understand how to work each of these problems.  You should begin
working on these problems in groups in recitation. You will probably
want to finish these problems outside of class. 
 If you have questions,
please ask your TA or instructor. If you find a problem difficult,
consider working similar problems from the text for additional practice.

\begin{enumerate}
\item Work the following problems from Stewart, section 1.5.
\#1, 3, 7, 9, 15, 16, 20, 23, 24.  

\item   Solve the following equations. Hint: You should be able to
work these in your head--don't use a calculator.  
\begin{enumerate}
\item  $2^x = 4$
\item $a^x = a $ (Assume that $a>0$.)
\item  $ a^x = 1$
\item  $4^x = 2$
\item  $8^x = 4$
\item   $4^x = 8$
\end{enumerate}


\item Suppose that you are extremely lucky and find a bank that pays
100\% interest. If this is simple interest, then you deposit \$1 and
at the end of the year, you receive your \$1 in principal and \$1 in
interest for a total of \$2. If the interest is compounded twice a
year, your account is contains $(1+ 1/2)^2 = \$2.25$ at the end of the
year.  
\begin{enumerate}
\item Suppose that interest is compounded 4 times a year, what is
the value of your account at the end of the year?  
\item Write a formula which gives the value of the count if the
interest is compounded $n$ times a year?  
Check your formula against the case $n=1, 2, 4$ considered above.

\item Use your answer to the previous part  to complete the following
table:
\begin{tabular}{r|rrrrrrr}
Number of compounding periods & 1 & 2 & 4 & 100 & 1,000& 100,000 \\
Value      of account         & 2 & 2.25 & 
\end{tabular}
What happens to the account as the number of compounding periods
increases?

\item Suppose now that we have a more reasonable interest of
10\%. What is the value of your account after 1 year if you pay
interest twice a year?  What if you pay interest 100,000 times a year?


\end{enumerate}



\item Are all exponential functions the ``same''?  More precisely, can
you obtain the graph of $  f(x ) = a^x$ from the graph of  $g(x) =
e^x$, by a vertical stretching?  By a horizontal stretching? Does it
matter if $a> 1$ or if $0<  a< 1$?  




\end{enumerate}
\end{flushleft}
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