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\title{Mathematics 226, Spring 2000}
\author{Midterm I}
\date{February 16, 2000}
\maketitle

\begin{enumerate}

\item Let $z=f(x,y)$ and $x = r \, \cos \theta$, $y = r \, \sin
\theta$.  Find $\displaystyle \frac{\partial z}{\partial r}$,
$\displaystyle \frac{\partial z}{\partial \theta}$, and $\displaystyle
\frac{\partial^2 z}{\partial r \partial \theta}$.
\hfill{(15)}

\item Sketch the level surfaces (level sets) of the function $f(x,y) =
e^{xy}$.
\hfill{(10)}

\item
\begin{enumerate}

\item
Find an equation for the plane tangent to the graph of
\[
  z = \ln(x^2+y^2)
\]
at the point where $(x,y) = (1,-1)$. 
\hfill{(15)}

\item
Show that the plane in part (a) is perpendicular to the line through
the points $(3,2,0)$ and $(4,1,-1)$.
\hfill{(10)}

\end{enumerate}

\item
\begin{enumerate}

\item
Describe and sketch the surface whose equation in spherical
coordinates is
\[
  \rho = 14 \, \cos \phi .
\]
\hfill{(15)}

\item Find an expression in rectangular coordinates for the equation
of part (a).  (You may assume that $0 \leq \phi \leq \frac\pi2$.)
\hfill{(10)}

\end{enumerate}

\item The curvature of a space-curve can be computed by
\[
  \kappa = \frac{|\boldsymbol{r}'(t) \times \boldsymbol{r}''(t)|}%
                {|\boldsymbol{r}'(t)|^3}.
\]
Use this formula to show that if the curve is the graph of a function
in the $x$-$y$ plane and $y=f(x)$, then
\[
  \kappa = \frac{|f''(x)|}{(1 + (f'(x))^2)^{3/2}}.
\]
\hfill{(10)}

\item A certain number $L$ is computed through the formula
\[
  L = \frac1{a-b} ,
\]
where $a=3.00$ and $b=2.50$.  Both numbers are accurate within the
given number of digits, but any further digits are unknown.  How many
digits of the final result can you trust? (Use differentials.  A
trial-and-error approach will get no credit!)
\hfill{(15)}

\end{enumerate}



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