\input epsf
\input calc
\def\problemspace{0.25in}

%% TITLE

\noindent
{\bf Math 225: Calculus III}\hfill Name:\hbox to 2.5truein{\hrulefill}
\medskip\noindent
{\bf Exam II}\quad{\sl October 31, 1991}\hfill Score:\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}


%1
%% PROBLEM
Determine which of the following shaded regions corresponds to the domain of the function \hfil\break \hbox{$f(x,y) = {\sqrt{1 - (x^2+y^2)}\over x y}$}.
%% CORRECT
{\epsfxsize=1in\epsfbox{exam2.0.1.eps}}
%% WRONG
{\epsfxsize=1in\epsfbox{exam2.0.2.eps}}
%% WRONG
{\epsfxsize=1in\epsfbox{exam2.0.3.eps}}
%% WRONG
{\epsfxsize=1in\epsfbox{exam2.0.4.eps}}
%% WRONG
{\epsfxsize=1in\epsfbox{exam2.0.5.eps}}


%2
%% PROBLEM
Determine which function below has level curves of the form:

\bigskip
\centerline{\epsfxsize=2.5in\epsfbox{exam2.1.eps}}

%% CORRECT
{$f(x,y) = {y\over 3x^2}$}
%% WRONG
{$f(x,y) = x^2 + 3 y^2$}
%% WRONG
{$f(x,y) = 3 x^2 y$}
%% WRONG
{$f(x,y) = x^2 - 3 y^2$}
%% WRONG
{$f(x,y) = y - 3x^2$}


%3
%% PROBLEM
Determine which function below has a graph like the following:

\bigskip
\centerline{\epsfxsize=3in\epsfbox{exam2.2.eps}}

%% CORRECT
{$f(x,y) = x^3 - x^2 - y^2$}
%% WRONG
{$f(x,y) = x - 1 - y$}
%% WRONG
{$f(x,y) = x^2 - x - y$}
%% WRONG
{$f(x,y) = y - x^2 + x$}
%% WRONG
{$f(x,y) = y^2 - x^3 + x$}


%4
%% PROBLEM
Compute the limit $\lim_{(x,y)\to(1,1)} [\ln(x^2 - y^2) - \ln(x - y)]$.

%% CORRECT
$\ln(2)$
%% WRONG
0
%% WRONG
1
%% WRONG
$\infty$
%% WRONG
{\sl does not exist}



%5
%% PROBLEM
If $f(x,y,z) = \sin(x y)e^{-z} + e^y + x^2$, compute $f_{xz}(1,\pi,0)$.

%% CORRECT
$\pi$
%% WRONG
0
%% WRONG
1
%% WRONG
2
%% WRONG
$-\pi/e$


%6
%% PROBLEM
Let $z = e^{-x^2 - y^2}$, $x = u v$, and $y = u^2 - v^2$. Compute ${\partial z \over \partial v}$.

%% CORRECT
$e^{-x^2-y^2}(-2ux + 4yv)$
%% WRONG
$e^{-u^2+v^2}(-2u^2v + 4v)$
%% WRONG
$e^{-x^2-y^2}(-2x - 2y)$
%% WRONG
$e^{-u^2v^2}(-2u^2v + 4v(u^2 - v^2))$
%% WRONG
$e^{-u^4 - v^4}(-2x - 2y)$



%7
%% PROBLEM
A particle moves along a smooth path in the plane. At time $t = 1$ the particle is at the point $(3, -4)$ and its velocity is $\v = 2\i + 3\j$. At what rate is the distance of the particle to the origin changing at that moment?

%% CORRECT
{\sl decreasing \rm 1.20}
%% WRONG
{\sl increasing \rm 1.20}
%% WRONG
{\sl decreasing \rm 3.74}
%% WRONG
{\sl increasing \rm 3.74}
%% WRONG
{\sl neither increasing nor decreasing at $t = 1$}




%8
%% PROBLEM
Find the derivative of
$f(x,y,z) = x^3 y - 2 y^2 z$ 
in the direction of the vector $\a = -\i + \j + 2\k$ at the point $(1,1,1)$.

%% CORRECT
$-4.08$
%% WRONG
$0.00$
%% WRONG
$-10.0$
%% WRONG
$-0.82$
%% WRONG
$2.96$


%9
%% PROBLEM
Compute the gradient of the function $f(x,y,z) = \cos(x y) - z^2$.

%% CORRECT
$-y\sin(xy)\i - x\sin(xy)\j - 2z\k$
%% WRONG
$-y\sin(xy)\i + x\cos(xy)\j + 2z\k$
%% WRONG
$-y\sin(xy) - x\sin(xy) - 2z$
%% WRONG
$-y\sin(xy) + x\cos(xy) + 2z\k$
%% WRONG
$-\sin(xy) -  2z\k$



%10
%% PROBLEM
Find the direction in which the function  $f(x,y) = y^3 x^2 + x$ increases most rapidly at the point $(2,1)$.

%% CORRECT
$5\i + 12\j$
%% WRONG
$3\i + 3\j$
%% WRONG
$4\i + 2\j$
%% WRONG
17
%% WRONG
13



%11
%% PROBLEM
Determine the equation of the plane tangent to the graph of $f(x,y) = 2x^4 - 3xy^2 + y^3$ at the point $(1,1,0)$.

%% CORRECT
$5x - 3y - z = 2$
%% WRONG
$x - 3y - z = -2$
%% WRONG
$8x - 6y - z = 2$
%% WRONG
$4x - 6y + z = -2$
%% WRONG
$5x - 3y = 2$



%12
%% PROBLEM
Find all the critical points of the function \hbox{$f(x,y) = 14x^3 - 21 x^2 y + 5 y^3 + 24 y + 1$}.

%% CORRECT
{$(-2,-2)$, $(2,2)$}
%% WRONG
{$(0,1.26)$, $(0,-1.26)$, $(-2,-2)$, $(2,2)$}
%% WRONG
{ $(-2,-2)$, $(-2,2)$, $(2,-2)$, $(2,2)$,}
%% WRONG
{$(0,1.26)$, $(0,-1.26)$, $(-2,2)$, $(2,-2)$}
%% WRONG
{$(0,1.26)$, $(0,-1.26)$}



%13
%% PROBLEM
Choose the statement below that is {\bf true} about the function \hbox{$f(x,y) = e^{-x}(x^2 + y^2)$}.

%% CORRECT
{\sl $f$ has a saddle point at $(2,0)$}
%% WRONG
{\sl $f$ has a local maximum at $(2,0)$}
%% WRONG
{\sl $f$ has a local minimum at $(2,0)$}
%% WRONG
{\sl $(2,0)$ is {\bf not} a critical point of $f$}
%% WRONG
{\sl none of the above}



%14
%% PROBLEM
Find the minimum of $f(x,y) = x^2 - 6 x y + y^3$ on the rectangle
\hbox{$0 \le x \le 20$}, \hbox{$0 \le y \le 10$}.

%% CORRECT
$-108$
%% WRONG
$-106$
%% WRONG
$-120$
%% WRONG
$-200$
%% WRONG
$0$



%15
%% PROBLEM
Find a point on the curve $13x^2 + 10 x y + 13 y^2 = 4$ where the value of the function $f(x,y) = x^2 y^2$ is greater than or equal to its value at any other point on the curve.

%% CORRECT
$({1\over2},-{1\over2})$
%% WRONG
$({1\over3},{1\over3})$
%% WRONG
$(-1,1)$
%% WRONG
$(1,-{1\over\sqrt{5}})$
%% WRONG
$({2\over\sqrt{13}}, -{2\over\sqrt{13}})$



%% END
\bye
