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\textbf{Math 126: Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Exam I}\quad \textsl{October 3, 2002}
\hfill\makebox[2.5in][r]{Section:\hrulefill}

\bigskip

There are 6 problems on 6 pages (including the cover page)
 worth a total of  100 points. 
%  You start
% with ?? points.
 Unless otherwise indicated, 
each part of a problem is worth the same number of points.

\medskip
You may use a calculator if you wish.  However, all answers must be
exact, e.g., 
\begin{math}
  1.414
\end{math}
is not equal to 
\begin{math}
  \sqrt 2.
\end{math}

\medskip
To receive partial credit on a problem, you must \emph{show your work}
including
\emph{all important steps}. No credit will be given for an answer if no
work is shown.

\bigskip

\

\bigskip

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 \problemspace



\begin{enumerate}

% \item (?? points) 
%   \begin{enumerate}
%   \item  Solve the equation $x^{\ln x}=e^4$

% \answerspace  


%    \item Use logarithmic differention to find the derivative of
% $$ y= \frac{(x+1)^2\tan x}{(x+3)^2}.$$

% \answerspace

%   \item Find the derivative of $x^{\ln x}$
%   \end{enumerate}


% \answerspace

%\problemspace

\item  (20 points)
Consider the function $f(x)= \frac{e^{3x} - e^{-3x}}2.$

  \begin{enumerate}

  \item Write 
    \begin{math}
      f
    \end{math}
in terms of hyperbolic trigonometric functions.

\answerspace

  \item Show that it is increasing for all $x$.

\answerspace

  \item What is its range?

\answerspace

  \item What is the range of its inverse function $g=f^{-1}$?

\answerspace

  \item Find $g'(a)$, where $a=0$.

\answerspace
  \end{enumerate}

\problemspace

\item  (20 points)
  \begin{enumerate}
  \item
  \label{parta}  Find $\lim_{x \to 0} \frac {\ln (1+2x)}{x}$.

\answerspace 

\item  Evaluate $\lim_{x \to 0} (1+2x)^{1/x}$.

\answerspace

  \end{enumerate}

\problemspace


% \item (?? points)

%  Find the equation of the tangent line to the curve
%  $$y=\sin^{-1}\frac{x}{2}$$
%  at the point $(\sqrt 2,\frac{\pi}{4})$.

%  \answerspace

 % \item (?? points) 
%  If $f(x)$ is a function whose derivative is given by
%  $$f'(x)= 1+ \frac{1}{\sqrt {1-x^2}}$$
%  and $f(\frac{1}{\sqrt 2})=\frac{\pi}{4}$, find $f(x)$.

%  \answerspace

 \item (20 points)
 Evaluate each of the following integrals.

 \begin{enumerate}
 % \item  $\int\limits_1^e \frac{x^2 +x+1}{x}dx$

%  \answerspace

 \item
 $\int\limits_0^{{\sqrt 3}/{2}} \frac{dx}{1+4x^2}$

 \answerspace


 % \item

%  $\int \frac{(\ln x)^2}{x}dx$


 


 

%  \item (?? points)
%  Find 
%  \begin{math}
%    \lim _{x\to \frac \pi 2} \left(\frac \pi 2 -x\right) \tan (x).
%  \end{math}

%  \answerspace 

%  \item (?? points)
%  Evaluate 

\item
 \begin{math}
   \int _3 ^6 \frac {5x-1}{x^2-x-2}\, dx.
 \end{math}
\end{enumerate}

 \answerspace

\problemspace

\item (20 points) Calculate the following integrals.
\begin{enumerate}
\item 
  \begin{math}
    \int \frac {\ln x}{x^3}\, dx.
  \end{math}

\answerspace

\item
  \begin{math}
    \int \frac {x^2}{\sqrt{1-x^2}}\, dx.
  \end{math}

\end{enumerate}
  
\answerspace 

% \problemspace

\problemspace


\item (10 points)
 A sum of money is invested at a fixed rate of interest,
compounded continuously. The investment doubles in ten years.
Find the interest rate.

\answerspace

\item (10 points)
If 
\begin{math}
  g
\end{math}
is a continuous function satisfying 
\begin{math}
  \int _1^{e^2} g(x)\,dx = 2002,
\end{math}
find 
\begin{math}
  \int _0 ^1 e^{2x}\,g(e^{2x})\,dx.
\end{math}
\answerspace

\end{enumerate}

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