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\title{Math 1b. Reviewing the Definite Integral}

\date{Spring 2006}

\begin{document}

\maketitle


\begin{enumerate}

\item
Which of the following are equal to 
$$
\int_1^5 \frac{\ln x}{x} \, dx?
$$
Please circle all of the correct answers.  You do not need to justify your solution.
\begin{enumerate}

\item
$\displaystyle \sum_{i = 1}^5 \frac{\ln x_i}{x_i} \Delta x$, where $\Delta x = 4/n$ and $x_i = 1 + i \Delta x_i$.

\item
$\displaystyle \lim_{n \to \infty} \sum_{i = 1}^5 \frac{\ln x_i}{x_i} \Delta x$, where $\Delta x = 4/n$ and $x_i = 1 + i \Delta x_i$.

\item
$\displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \frac{\ln x_i}{x_i} \Delta x$, where $\Delta x = 4/n$ and $x_i = 1 + i \Delta x_i$.

\item
$(\ln 5)^2/2$

\item
$\displaystyle \frac{1}{2} \left( \frac{\ln 5}{5}\right)^2 - \frac{1}{2} ( \ln 1)^2$

\item
$\displaystyle \frac{1}{5^2} - \frac{1}{1^2}$


\item
$\ln(\ln( 5)) - \ln(\ln 1))$

\item
$\displaystyle \frac{\ln 1}{1} + \frac{\ln 2}{2} + \frac{\ln 3}{3} + \frac{\ln 4}{4}$

\end{enumerate}

\clearpage

\item
Put the following in {\em ascending\/} order (with ``$=$'' or ``$<$'' signs between each expression.  You do not need to justify your solution.  [{\em Hint}: Think about which expressions are positive, which are negative, and which are zero.  A picture may be helpful.]  
\begin{enumerate}

\item
$\displaystyle \int_2^6 \ln t \, dt$

\item
$\ln 2 + \ln 3 + \ln 4 + \ln 5$


\item
$\ln 3 + \ln 4 + \ln 5 + \ln 6$

\item
zero

\item
$\ln(2/6)$


\item
$\displaystyle \lim_{h \to 0} \frac{\ln(2 + h) - \ln 2}{h}$

\end{enumerate}



\end{enumerate}


 \end{document}