%First version of exam 3

\documentclass[12pt]{letter}
\usepackage{amsmath}
\textwidth 6.5in\oddsidemargin 0in
\textheight 9.5in\topmargin -1.0in

\def \sm {\smallskip\noindent}
\def \bm {\bigskip\noindent}
\def \mm {\medskip\noindent}
\def \hs {~~~~~~}
\def \hh {~~~}
\def \ds {\displaystyle}
\def \pfx {\frac{\partial f}{\partial x}}
\def \pfxx {\frac{\partial^2f}{\partial x^2}}
\def \pfy {\frac{\partial f}{\partial y}}
\def \pfyy {\frac{\partial^2f}{\partial y^2}}
\def \pfxy {\frac{\partial^2f}{\partial x\partial y}}
\def \pfyx {\frac{\partial^2f}{\partial y\partial x}}

\begin{document}
\centerline{\bf MULTIPLE CHOICE SECTION - EACH PROBLEM IS WORTH  5 POINTS }
\centerline{\bf (In Problems 1-10 mark the correct answer on the front cover)}

\bm
{\bf 1.} A cube having volume of 64 units sits in 3-space with each edge
parallel to an
axis.  The center of one of the sides of the cube is at the point
$(1,1,1)$.  Which of the
following points could be a corner of the cube?

\mm
(a) $(-1,1,1)$ \hfil (b) $(-1,3,3)$ \hfil (c) $(-3,3,5)$ \hfil (d)
$(3,1,-1)$ \hfil
(e) $(5,1,-3)$

\vskip2.0in
{\bf 2.}  Suppose that $f(x,y)$ is a function with $\ds\pfx = 3x^2 + e^{xy}
- y$. Find
$\ds\pfxx$.

\mm
(a) $6 + y^2e^{xy}$ \hfil (b) $xe^{xy} - 1$ \hfil (c) $6x + xe^{xy} - 1$
 \hfil (d) $6x + e^{xy}$ \hfil (e) $6x + ye^{xy}$


\vskip2.0in
{\bf 3.}  Suppose that $f(x,y)$ and $g(x,y)$ are both linear functions.
Which of the
following must also be linear functions:

\sm
I. $f(3x,y)$ \hfil II. $f(x,y)g(x,y)$ \hfil III. $2f(x,y) - g(x,y)$ \hfil
IV. $xf(x,y)$ \hfil V. $g(x,xy)$

\mm
(a) I and III \hfil (b) only IV \hfil (c) II and VI \hfil (d) only III
\hfil (e) III and V

\newpage
{\bf 4.} Widgets-R-Us makes two kinds of widgets: type X and type Y.  The
profit from
selling $x$ widgets of type X and $y$ widgets of type Y is a function
$P(x,y)$.  Suppose
that
$$
P(100,150) = \$10,000,\hh \frac{\partial P}{\partial x}(100,150) = 200,\hh
\text{and} \hh
\frac{\partial P}{\partial y}(100,150) = -100.
$$
Estimate the profit the company will earn if it sells 120 widgets of type X
and 160
widgets of type Y.

\mm
(a) \$11,000 \hfil (b) \$13,000 \hfil (c) \$12,000 \hfil (d) \$15,000 \hfil
(e) \$14,000

\vskip1.5in
{\bf 5.} For what value(s) of $k$ will the following system of equations
have infinitely
many solutions:
$$\aligned
x + y + z &= 3\\
x + 2y + 3z &= 6\\
-2x + ky -2z &= -4
\endaligned$$

\mm
(a) $k = -2$ \hfil (b) $k = 2, 4$ \hfil (c) no values of $k$ \hfil (d) all
$k \neq -2$
\hfil (e) $k = 1$

\vskip2.0in
{\bf 6.} Which of the following is the function which must be minimized in
order
to find the Least Squares Fit straight line ($y = ax + b$) through the three
points $(0,3)$, $(1,1)$, $(4,2)$?

\mm
(a) $E = (3a + b)^2 + (a + b - 1)^2 + (2a + b - 4)^2$\\
(b) $E = (b - 3)^2 + (a + b - 1)^2 + (4a + b - 2)^2$\\
(c) $E = (a + b)^2 + (a + b - 3)^2 + (a + b - 2)^2$\\
(d) $E = (b - 3)^2 + (a + b -1)^2 + (2a + b - 4)^2$\\
(e) $E = (a + b - 1)^2 + (2a + b - 2)^2 + (3a + b - 2)^2$

\newpage
{\bf 7.} In 3-dimensions a line can be described by three equations such as
the following:
$$x = 1 + t,\ y = 2,\ z = 3,$$
where $t$ is allowed to be any real number. i.e.\ for each value of $t$, we
get a point
$(1 + t,2,3)$ on the line.  Find the value(s) of $t$ for which the
corresponding point on
the line is at a distance of $\sqrt{5}$ units from the point $(1,2,2)$.

\mm
(a) $t = -3, 1$ \hfil (b) $t = -1$ \hfil (c) $t = -2, 2$ \hfil (d) $t =
-\sqrt{24},
\sqrt{24}$ \hfil (e) $t = 3$

\vskip2.0in
{\bf 8.}  Find the equation of the tangent plane to the graph of the function
$f(x,y)= - 2xy$ at the point $(1,-1)$.

\mm
(a)  $z= 2x - 2y + 2$ \hfil  (b)  $z= -2x + 2y + 2$ \hfil (c)  $z= -2x - 2y
+ 2$

(d) $z = 2x - 2y -2$ \hs\hs (e) $z = 2x + 2y + 2$

\vskip2.0in
{\bf 9.}  A function $f(x,y)$ satisfies $\ds\pfx = 0 = \pfy$ at the point
$(7,10)$.
Suppose also that $\ds\pfxx = -3$, $\ds\pfyy = 4$, and $\ds\pfxy = -4$ at
the point
$(7,10)$.  Which of the following is true?

\mm
(a) $(7,10)$ is a relative maximum point.\\
(b) $(7,10)$ is a saddle point.\\
(c) $(7,10)$ is a relative minimum point.\\
(d) $(7,10)$ is not a critical point.\\
(e) $(7,10)$ is a critical point, but the second derivative test is
inconclusive.

\newpage
{\bf 10.} Write the equation of the plane which contains the points $(-2,0,0)$,
$(0,3,0)$, and $(0,0,6)$.

\mm
(a) $z = 3x - 2y + 6$ \hfil (b) $z = -\frac13x + \frac12y + 3$ \hfil
(c) $z = \frac12x - \frac13y - 2$

(d) $z = -3x + 2y + 6$ \quad\quad\hh\  (e) $z = -2x + 3y + 6$

\vskip2.5in
\centerline{\bf PARTIAL CREDIT PART}
\centerline{\bf (In Problems 11-15 show all work on the paper)}
\bigskip

\noindent
{\bf 11.}   (8 points)
Consider the function $f(x,y) = 3x^2 + 3y^2$. Sketch the level curves for
$f(x,y)$ at the
values $z = -3, 0, 3, 9$ on the following graph. Label each curve.  If a
level curve
doesn't exist, say so and explain why.

\newpage
{\bf 12.} (7 points) The profit from selling $x$ units of one item and $y$
units of another
item is given by $P(x,y) = 3x^4 - x^2y + xy^2 - y$.  Assuming the company
will produce a
total of 100 items, use the Method of Lagrange multipliers to set up the
system of equations
whose solution gives the possible points at which the profit could be
maximized.  You do
{\bf not} have to solve the system.

\vskip3.2in
{\bf 13.} (10 points) Find the linear function $f(x,y)$ which satisfies
$f(1,1) = 5$,
$f(0,1) = 3$, and $f(1,2) = 6$.

\newpage
{\bf 14.} (10 points) Identify all pairs $(x,y)$ which satisfy the
following system of
equations:
$$\aligned
2y &= 4x\lambda\\
x &= 2y\lambda\\
2x^2 + y^2 &= 12
\endaligned$$
You do {\bf not} need to find the corresponding value(s) of $\lambda$.

\newpage
{\bf 15.} (15 points) From the Method of Least squares, the regression line
$y = ax + b$
which best fits the three points $(0,2)$, $(1,-1)$, and $(3,0)$ can be
determined by
finding the point $(a,b)$ where the function
$$
E(a,b) = 5a^2 + 6ab + 3b^2 + 2a - 2b + 5
$$
is a minimum.

(a) Find all the critical point(s) of $E(a,b)$.

\vskip4.0in
(b) Apply the second derivative test to each critical point to determine
whether it is a
relative maximum, relative minimum, or saddle point.

\end{document}

