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\begin{document}

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

\textbf{Exam I}\quad \emph{September 30, 1999}

\bigskip
There are 7 problems worth of total of 80 points. You start with 20
points. To receive full credit you must show all your work and include
all important steps.

\bigskip

\begin{enumerate}

\item (10 pts) Inverse Functions.

  \begin{enumerate}[a)]

  \item
  Find the largest interval on the $x$-axis containing the number $x=1$
  on which the function \[ f(x) = x + \frac{1}{4x} \] is one-to-one.

  \answerspace

  \item Find the inverse $f^{-1}(x)$ of the function $f(x)$ restricted
  to the interval determined in a).

  \answerspace

  \end{enumerate}

%\problemspace
%
%\item (10 pts) Define the following functions.
%
%  \begin{enumerate}[a)]
%
%  \item $\ln(x)$
%
%  \answerspace
%
%  \item $e^x$
%
%  \answerspace
%
%  \item $a^x$ for a real constant $a > 0$.
%
%  \answerspace
%
%  \item $\log_a(x)$ for a real constant $a > 0$.
%
%  \answerspace
%
%  \end{enumerate}
 
\problemspace

\item (10 pts)
Let $f(x) = x^{1/x}$ for $x > 0$.

  \begin{enumerate}[a)]
  \item Compute $f'(x)$.

  \answerspace

  \item Compute $\lim_{x\to \infty} f(x)$.

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts)
An ice cube tray is filled with water at $70^\circ$F and placed in
a freezer kept at $-15^\circ$F. Ten minutes later the water is at
$60^\circ$F. Use Newton's Law of cooling to determine when the
water will begin to freeze (reach $32^\circ$F). Recall that Newton's Law
of Cooling is
   \[
   \frac{d}{dt}(T(t) - T_S) = k(T(t) - T_S)
   \]
where $T(t)$ is the temperature of an object and $T_S$ is the constant
temperature of the surrounding medium.

\problemspace

\item (10 pts) Compute the following limits.

  \begin{enumerate}[a)]

  \item $\lim_{x\to 0} \frac{e^x - (x+1)}{x^2} $

  \answerspace

  \item $\lim_{x\to \pi^-} \Bigg(\frac{1}{\sin(x)} + \frac{1}{x-\pi}\Bigg)$

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) Rates of Growth.

  \begin{enumerate}[a)]

  \item Show that $\ln(\ln(x)) = o(\ln(x))$ as $x\to \infty$.

  \answerspace

  \item Show that $100e^x + x^2$ grows at the same rate as $e^x + x$ as $x \to
  \infty$.

  \answerspace

  \end{enumerate}

\problemspace

\item (20 pts) Compute the following integrals.

  \begin{enumerate}[a)]

  \item $\int_1^4 \frac{1}{\sqrt{x} + x}\, dx$

  \answerspace

  \item $\int \frac{dx}{x^2 + 2x + 5}$

  \answerspace

\newpage

  \item $\int \frac{e^x dx}{\sqrt{4e^{2x}-1}}$

  \answerspace

  \item $\int_0^{\pi/2} x^2\cos(x)\,dx$

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) Solve the initial value problem
\[
\frac{dy}{dx} = x + xy\qquad y(0) = 2
\]

\end{enumerate}

\end{document}
