\input epsf
\input calc
\def\problemspace{0.2in}

%% TITLE
\noindent
{\bf Math 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\quad{\sl October 29, 1992}\hfill
Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this answer sheet. There are 15 multiple
choice questions worth 6 points each. You start with 10 points.

\def\answerspace{0.25in}


%% PROBLEM
Let $f(x,y,z) = {\ln(xy^2)\over z^3}$. Compute $f_{zx}(1,2,3)$.
%% CORRECT
$-{1\over27}$
%% WRONG
$-{2\over\ln(2) 81}$
%% WRONG
$0$
%% WRONG
$-{2\over\ln(2) 27}$
%% WRONG
$-{1\over3}$



%% PROBLEM
Compute the limit $\lim_{(x,y)\to(0,0)} {xy^3\over x^4 + y^4}$.
%% CORRECT
{{\sl does not exist}}
%% WRONG
${3\over8}$
%% WRONG
${1\over8}$
%% WRONG
$\infty$
%% WRONG
$0$


%% PROBLEM
Calculate the gradient of the function $f(x,y,z) = x^2y - z^2x + yz$.
%% CORRECT
$(2xy - z^2)\i + (x^2 + z)\j +  (y - 2xz)\k$
%% WRONG
$2xy - z^2 + x^2 + z + y - 2xz$
%% WRONG
$2xy - z^2 + y + z$
%% WRONG
$x^2y\i - z^2x\j + yz\k$
%% WRONG
$2xy\i - z^2\j + (y + z)\k$


%% PROBLEM
Find the direction in which the function $f(x,y) = xy - x/y$ is increasing
most rapidly at the point $(5,3)$.
%% CORRECT
${8\over3}\i + {50\over9}\j$
%% WRONG
$3\i - {5\over 9}\j$
%% WRONG
${74\over 9}$
%% WRONG
${24\over5}\i + {78\over25}\j$
%% WRONG
${40\over3}$

%% PROBLEM
Calculate the derivative of $f(x,y,z) = xy  - z^2$ in the direction of
$\i - \j + 2\k$ at the point $(3,4,1)$.
%% CORRECT
$-{3\over\sqrt{6}}$
%% WRONG
$4\i + 3\j - 2\k$
%% WRONG
$-3$
%% WRONG
$1$
%% WRONG
${4\over\sqrt{6}}\i - {3\over\sqrt{6}}\j - {4\over\sqrt{6}}\k$


%% PROBLEM
Determine the critical points of the function $g(u,v) = u^2 v - 27v^3 - 2u$.
%% CORRECT
$(3,{1\over3}), (-3,-{1\over3})$
%% WRONG
$(3,{1\over3}), (-3,{1\over3}), (3,-{1\over3}), (-3,-{1\over3})$
%% WRONG
$(9,{1\over9}), (-9,-{1\over9}), (0,{1\over9}), (0,-{1\over9})$
%% WRONG
$(0,0), (1,1), (-1,-1)$
%% WRONG
$(1,1), (9,1), (-1,-1), (-9,-1)$


%% PROBLEM
Which of the following statements is true about the function
$f(x,y) = e^{x^2 - 3xy + y^2}$.
%% WRONG
{$f(x,y)$ has a critical point at $(1,-1)$}
%% WRONG
{$f(x,y)$ has a local maximum at $(0,0)$}
%% WRONG
{$f(x,y)$ has a local minimum at $(0,0)$}
%% CORRECT
{$f(x,y)$ has a saddle point at $(0,0)$}
%% WRONG
{{\sl none of the above}}
%% STET

%% PROBLEM
Find the minimum value of the function $f(x,y) = x^2 - 2x y + y - x$ in the square region $0 \le x \le 2$, $-1 \le y \le 1$.
%% CORRECT
$-{5\over4}$
%% WRONG
$-{1\over4}$
%% WRONG
$-2$
%% WRONG
$-1$
%% WRONG
$-{1\over2}$


%% PROBLEM
Determine the maximum value of the function $f(x,y) = x - 2 y$ subject to the constraint $x^2 + 4y^2 = 4$.

%% CORRECT
$2.828$
%% WRONG
$1.000$
%% WRONG
$1.436$
%% WRONG
$2.000$
%% WRONG
$3.936$


%% PROBLEM
Which of the following sets of equations must be solved to find the extreme values of the function $f(x,y) =  x^4 y + x^2 y + x y^3 $ subject to the constraint $x^4 + 4xy + y^4 = 16$

%% CORRECT
{\vtop{
\hbox{$4x^3y + 2xy + y^3 = \lambda(4x^3 + 4y)$}
\hbox{$x^4 + x^2 +3xy^2 = \lambda(4x + 4y^3)$}
\hbox{$x^4 + 4xy + y^4 = 16$}
\hbox{\strut}
}}
%% WRONG
{\vtop{
\hbox{$4x^3y + 2xy + y^3 = \lambda(4x^3 + 4y)$}
\hbox{$x^4 + x^2 +3xy^2 = \lambda(4x + 4y^3)$}
\hbox{$4x^3 + 4y + 4y^3 = 0$}
\hbox{\strut}
}}
%% WRONG
{\vtop{
\hbox{$4x^3 + 2xy + y^3 = \lambda(4x^3 + 4y^3)$}
\hbox{$x^2 +3xy^2 = \lambda(4x + 4y^3)$}
\hbox{\strut}
}}
%% WRONG
{\vtop{
\hbox{$4x^3 + 2xy = \lambda(4x^3 + 4y)$}
\hbox{$x^2 +3xy^2 = \lambda(4x + 4y^3)$}
\hbox{$x^4 + 4xy + y^4 = 16$}
\hbox{\strut}
}}
%% WRONG
{\vtop{
\hbox{$4x^3 + x^4 + 2xy + x^2 + 3y^2 = \lambda(4x^3 + 4y)$}
\hbox{$x^4 + 4x^3 y + x^2 +3xy^2 = \lambda(4x + 4y^3)$}
\hbox{$x^3 + x + y + y^3 = 4$}
\hbox{\strut}
}}


%% PROBLEM
Find the equation of the plane tangent to the graph of
$g(x,y) = x^3y - y^2 + x$ at the point $(1,1,1)$.
%% CORRECT
$4x - y - z = 2$
%% WRONG
$4x - y = 3$
%% WRONG
$x + y + z = 3$
%% WRONG
$x + y = 2$
%% WRONG
$(1 + 3x^2y) + (x^3 - 2y) + z = 4$


%% PROBLEM
Let $z = x^2 \cos(y + x)$. Suppose $x = g(u,v)$, $y = h(u,v)$ and at the point $(u,v) = (0,0)$, $g(0,0) = \pi$, $h(0,0) = 0$, and:
$$\d{x}/du = 3,\quad \d{x}/dv = -1,\quad \d{y}/du = -2,\quad \d{y}/dv = -2$$
Compute $\d{z}/du$ at $(u,v) = (0,0)$.

%% CORRECT
$-6\pi$
%% WRONG
$0$
%% WRONG
$4\pi$
%% WRONG
$-2\pi$
%% WRONG
$1$


%% PROBLEM
A certain type of salt crystal is being grown in a chemistry lab.
The crystal is the shape of a rectangular box. After 1 hour, the crystal is observed to be 1~mm wide, 2~mm deep, and 2.5~mm high with the width, depth, and height increasing at a rate of 0.2~mm, 0.4~mm, and 0.5~mm per hour, respectively. How fast is the volume of the crystal changing at that moment.
%% CORRECT
$3.0~{\rm mm}^3$ per hour
%% WRONG
$2.25~{\rm mm}^3$ per hour
%% WRONG
$1.8~{\rm mm}^3$ per hour
%% WRONG
$1.1~{\rm mm}^3$ per hour
%% WRONG
$2.75~{\rm mm}^3$ per hour


%% PROBLEM
Determine which of the following plots represents the level curves of the function $f(x,y) = 2x^2 + 4y^2$.
%% CORRECT
\epsfxsize=1.5truein\epsfbox{exam2.1.1.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.1.2.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.1.3.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.1.4.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.1.5.eps}


%% PROBLEM
Which of the following plots represents the graph of the function
$f(x,y) = 2y^3 + 3x^2 - 6xy$
%% CORRECT
\epsfxsize=1.5truein\epsfbox{exam2.2.1.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.2.2.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.2.3.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.2.4.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.2.5.eps}

%% END
\end
