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\textbf{Math 126: Calculus II}
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\textbf{Exam I}\quad \textsl{September 28, 2000}
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\bigskip

There are 6 problems on 6 pages worth a total of 84 points.  You start
with 16 points. Each part of a problem is worth the same number of points.

\medskip
You may use a calculator if you wish.

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

\bigskip

\begin{enumerate}

\item (18 pts) Find the derivatives.

\begin{enumerate}[a)]

\item
Let $f(x) = x\ln x$. Find the value of $(f^{-1})'(2e^2)$.
[Hint: $f(e^2) = 2e^2$.]

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\item  $\frac{d}{dx}\cos^{-1}(x^{-1})$.

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\item $\frac{d}{dx}x^{\cosh(x)}$.

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\end{enumerate}

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\item (18 pts) Compute the integrals.

  \begin{enumerate}[a)]

  \item $\int e^{x}\sin(1 + e^{x})dx$.

  \answerspace

  \item $\int_{0}^{1}\frac{x}{1+x^4}\,dx$

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  \item $\int \frac{x}{x+1}\,dx$

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  \end{enumerate}

\problemspace

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\item (12 pts)  Compute the limits. 

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\begin{enumerate}[a)]
\item $\lim_{x\to 0} \frac{\sin^{-1}(x)}{x}$

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\item $\lim _{x\to 0^+} e^{-1/x}\ln(x)$

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\end{enumerate}

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\item (12 pts) 
In each part of this problem you are presented with two functions
$f(x)$ and $g(x)$ and you are to decide which of the following three
possibilities holds as $x\to\infty$.

\begin{itemize}

\item[i)] $g(x)$ grows faster than $f(x)$.

\item[ii)] $f(x)$ grows faster than $g(x)$.

\item[iii)] $f(x)$ and $g(x)$ grow at the same rate.

\end{itemize}

Write the answer as i), ii), or iii). As always, you must show all work
justifying your answer.

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\begin{enumerate}[a)]

\item $f(x)=\frac{1}{x}$ and $g(x) = \sin (\tan^{-1}(x))$.

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\item $f(x)=\sinh(\ln(x))$ and $g(x) = x$.

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\end{enumerate}

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\item (12 pts) Warfarin is a drug used as an anticoagulant.  After
   administration of the drug ends, the quantity remaining in a
   patient's body decreases at a rate proportional to the quantity
   remaining.  The half-life of Warfarin in the body is 37 hours.  How
   many days does it take for the drug level in the body to be reduced
   to 10\% of the original level.  

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\item (12 pts)

  \begin{enumerate}[a)]

  \item Solve the differential equation\quad $x y' + 2 y = x$.

  \answerspace

  \item Solve the initial value problem\quad $x^2 y' = y$,\quad $y(1) = 1$.

  \answerspace

  \end{enumerate}

\end{enumerate}

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