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\begin{document}

\title{Mathematics 2D, Fall 1997}
\author{Marcel Oliver}
\date{Final Exam}

\maketitle

\begin{enumerate}
\item Locate and identify the extrema of
\[
  f(x,y) = x^2 -x + y^2 + 2y .
\]
\hfill (5)
\vspace*{\fill}

\item Find the point on the ellipse
\[
  x^2 + 4 y^2 = 4
\]
closest to $(1,0)$.
\hfill (10)
\vspace*{\fill}


\item Let
\[
  f(x,y) = x^3 \, y^2 .
\]
\begin{enumerate}
\item What is the largest directional derivative of $f$ at $(1,1)$?
\item In what direction is the derivative in part (a)?
\item Find the directional derivative of $f$ at $(1,1)$ in the
direction $(-1,-1)$.
\item What is the equation of the level curve of $f$ that passes
through the point $(1,1)$?
\item Show that the gradient of $f$ evaluated at $(1,1)$ is
perpendicular to the tangent line to the level curve there.
\end{enumerate}
\hfill (5+5+5+5+5)



\item Consider the part of the unit disk, centered at the origin,
which is located to the right of the $y$-axis.  Compute its center of
mass, assuming constant density.
\mbox{~~~~~~} \hfill (10)



\item Evaluate the following integrals.
\begin{enumerate}
\item $\displaystyle \iint_D \frac1{(x^2+y^2+1)^2} \, d A$

where $D$ is the circular disk with radius $a$ centered at the
origin.

\item $\displaystyle \int_0^{\tfrac\pi2} \int_y^{\tfrac\pi2}
  \frac{\sin x}{x} \, dx \, dy$
  
\item $\displaystyle \int_0^1 \int_{\sqrt{y}}^1
  \frac3{x^3+1} \, dx \, dy $

\end{enumerate}
\hfill (5+5+5)



\item The area element in polar coordinates is given by
\begin{equation}
  dA = r \, d \theta \, dr . \tag{*}
\end{equation}
\begin{enumerate}
\item Give a geometric justification of formula (*).

\item Derive (*) by using the transformation formula for double
integrals. 

\item Evaluate the integral
\[
  \int_{-\infty}^\infty \int_{-\infty}^\infty
  e^{-x^2-y^2} \, dx \, dy.
\]
\end{enumerate}
\hfill (5+5+5)



\item Use a suitable change of variables to find the area of the
region bounded by the graphs of
\begin{align*}
  \frac{x}{y} & = 1 & \frac{x}{y} & = 2 \\
  xy & = 1 & xy & = 2,
\end{align*}
where $x$ and $y$ are positive.
\hfill (10)



\item In a simple model the earth's atmosphere is assumed to extend to
an infinite height, while its density $\rho$ falls off exponentially.
I.e.,
\[
  \rho(r) = \rho_0 \, e^{-\alpha(r-R)}
\]
where $r$ is the distance to the center of the earth, $R$ is the
radius of the earth, $\alpha$ is a given constant, and $\rho_0$ is the
density at ground level (i.e.\ when $r=R$).

Use triple integration in spherical coordinates to show that the total
mass of the atmosphere is given by the expression
\[
  m = 4 \pi \, {\rho_0}
      \left(
        \frac{R^2}\alpha + \frac{2R}{\alpha^2} +  \frac2{\alpha^3}
      \right) .
\]
\hfill (10)


\end{enumerate}


\end{document}
