
\documentclass[12pt]{article}

 \newcommand{\examtitle}{{\bf\Large \centerline{Calculus 223 First Midterm Exam}  \par
                                 \centerline{\large Thursday February 14, 1996}}}

\newcounter{examprobnum}
\newcounter{pointTotal}
\newcommand{\prob}[1]{
   \stepcounter{examprobnum} \par
    \addtocounter{pointTotal}{#1}
    \noindent{\bf\Roman{examprobnum}. }(#1\%)
  }
\newcommand{\ds}{\displaystyle}
\newcommand{\NextPage}{\par\bigskip\par}
\begin{document}

\examtitle

\bigskip
\prob{14} (a) Find an equation for the tangent plane
to the surface $z=-11+x^2+y^3$  at the point $(x,y,z)=(4,0,5)$.


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\noindent(b) Find an equation for the tangent plane to
the surface $x+2y+3z-\cos(xyz)=18$ at the point $(x,y,z)=(4,0,5)$.


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\NextPage

\prob{10} (a) Evaluate
$\ds \lim_{(x,y)\to(0,0)} \tan^{-1} \frac{1}{x^2+y^2}$
or show that the limit does not exist.

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\noindent(b) Evaluate
$\ds \lim_{(x,y)\to(0,0)} \tan^{-1} \frac{xy}{x^2+y^2}$
or show that the limit does not exist.

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\NextPage

\prob{15} (a) Find the derivative of $f(x,y,z) = x^2+xy +xyz$
at the point $(1,2,3)$ in the direction
$$
   {\bf u} = \frac{2}{3}{\bf i} -\frac{1}{3}{\bf j}-  \frac{2}{3}{\bf k}.
$$

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\noindent(b) In what direction is this function increasing the fastest
at the point $(1,2,3)$?


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\NextPage

\prob{10} Find the quadratic polynomial which best approximates
the function  $f(x,y) = x^3+x^2y+y^3$ near the point $(x,y)=(1,2)$.

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\prob{15} Let $f(x,y) = 2x^2+2xy+y^2-8x-6y$.

\medskip

\noindent(a) What is the smallest   value $f(x,y)$ can take?

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%\vspace*{1in}

\noindent(b) What is the largest   value $f(x,y)$ can take?

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\prob{15} Minimize the function $f(x,y) = x^2-5xy+y^2$
on the square  $-1\le x\le 1$, $-1\le y\le 1$.


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\prob{15} A is curve given parametrically by the equations
$$
    x=4+t, \qquad y = u(t), \qquad z = v(t).
$$
The curve lies in both of the surfaces
$$
   z=-11+x^2+y^3, \qquad x+2y+3z-\cos(xyz)=18,
$$
of problem~I and passes through the point $P_0=(4,0,5)$ at
$t=0$.  Find the velocity vector of this curve at $t=0$.

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\prob{6} (True or false: Circle your choice.
One point for each correct answer,
-2 points for each incorrect answer.)

\medskip

Consider the function $f(x,y) = x^2+3xy+2y^2$.


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\noindent{\bf True False } The level curves of $f$ are ellipses.

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\noindent{\bf True False } The level curves of $f$ are hyperbolas.

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\noindent{\bf True False } The function $f$ takes only
values which are greater than or equal to zero.

\bigskip

\noindent{\bf True False } The function $f$ takes all values.
\end{document}
