\magnification=\magstep1
\input epsf
\input calc
\def\problemspace{0.25truein}

%% TITLE

\noindent
{\bf Math 225: Calculus III}\hfill Name:\hbox to 2.5truein{\hrulefill}
\medskip\noindent
{\bf Exam II}\quad{\sl March 21, 1996}\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 5 points each. You start
with 25~points.

\def\answerspace{0.25truein}


%1
%% PROBLEM
Find parametric equations of the line perpendicular to the surface defined by
$x^2 y - z^2 = 1$ at the point $(1,2,1)$.
%% CORRECT
$x = 1 + 4t$, $y = 2 + t$, $z = 1 - 2t$
%% WRONG
$x = 1 + 2x y t$, $y = 2 + x^2 t$, $z = 1 - 2 z t$
%% WRONG
$x = 1 + t$, $y = 2 + 2t$, $z = 1 - t$
%% WRONG
$x = t$, $y = 2t$, $z =  - t$
%% WRONG
$x = 1 - 4t$, $y = 2 + t$, $z = 1 + t$


%2
%% PROBLEM
Determine which function below has a graph like the following:

\bigskip
\centerline{\epsfxsize=2truein\epsfbox{exam2.1.eps}}

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


%3
%% PROBLEM
Compute the slope of the line tangent to the plane curve defined by the
equation\hfil\break
$ x^2 y^3 - x + y^2 = 1$ at the point $(1,1)$.
%% CORRECT
$-1/5$
%% WRONG
$1/4$
%% WRONG
$6$
%% WRONG
$1$
%% WRONG
$-1/6$


%4
%% PROBLEM
If $f(x,y,z) = \cos(x^2 y)e^{-z} + e^{y^2}x^3$, compute $f_{xz}(1,\pi/2,0)$.

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


%5
%% PROBLEM
Suppose $f(x,y)$ satisfies $f_x(6,13) = -4$ and $f_y(6,13) = 3/2$.
If $x = u v$, and $y = u^2 + v^2$, compute ${\partial f \over \partial v}$
at the point $(u,v) = (3,2)$.

%% CORRECT
$-6$
%% WRONG
$5$
%% WRONG
$-9$
%% WRONG
$1$
%% WRONG
$8$


%6
%% PROBLEM
Find the derivative of $f(x,y,z) = y \log(x+3) - 2xz^2e^y$
at the point $(-2,0,3)$ in the direction of
the vector $\a = -\i - 2\j + 2\k$.

%% CORRECT
$-2$
%% WRONG
$-6$
%% WRONG
$(18,36,24)$
%% WRONG
$(-18,-72,48)$
%% WRONG
$-3$


%7
%% PROBLEM
Compute the gradient of the function $f(x,y,z) = \sin(x y) + y^3 - xz^2$
at the point $(\pi,1,2)$.

%% CORRECT
$-5\i + (3-\pi)\j-4\pi\k$
%% WRONG
$-\i + 3\j - 4\pi\k$
%% WRONG
$-(1+4\pi)\i + 2\j - 4\k$
%% WRONG
$-\i + 3\j - (2-\pi)\k$
%% WRONG
$-5\i - \pi\j - 4\k$


%8
%% PROBLEM
Determine the equation of the plane tangent to the graph of $f(x,y) =
x^3 - 3x^2y +2y^4$ at the point $(1,1,0)$.

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


%9
%% PROBLEM
Find all the critical points of the function
$f(x,y) = 5 x^3 - 21 x y^2 + 14y^3 +  24 x + 3$.

%% CORRECT
{$(-2,-2)$, $(2,2)$}
%% WRONG
{$(1.36,0)$, $(-1.36,0)$, $(-2,-2)$, $(2,2)$}
%% WRONG
{ $(-2,-2)$, $(-2,2)$, $(2,-2)$, $(2,2)$,}
%% WRONG
{$(1.36,0)$, $(-1.36,0)$, $(-2,2)$, $(2,-2)$}
%% WRONG
{$(1.36,0)$, $(-1.36,0)$}


%10
%% PROBLEM
Choose the statement below that is {\bf true} about the function
$f(x,y) = e^{-y}(x^2 + y^2)$.

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

%% STET


%11
%% PROBLEM
Find the minimum of $f(x,y) = x^3 - 6 x y + y^2$ on
the rectangle \hbox{$0 \le x \le 10$}, \hbox{$0 \le y \le 20$}.

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



%12
%% PROBLEM
Find a point on the curve $13x^2 + 10 x y + 13 y^2 = 4$ where the value
of the function\hfil\break
$f(x,y) = x^2 y^2$ 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
$({2\over\sqrt{13}}, -{2\over\sqrt{13}})$


%13
%% PROBLEM
Compute $\int_0^1\int_0^y e^{y^2} \,dx\,dy$.
%% CORRECT
$(e-1)/2$
%% WRONG
$e$
%% WRONG
$0$
%% WRONG
${1\over2} - e$
%% WRONG
$1$


%14
%% PROBLEM
Reverse the order of integration in the integral $\int_1^2\int_1^{y^3}
f(x,y)\,dx\,dy$
%% CORRECT
$\int_1^8\int_{x^{1/3}}^2f(x,y)\,dy\,dx$
%% WRONG
$\int_1^{y^3}\int_1^2f(x,y)\,dy\,dx$
%% WRONG
$\int_1^2\int_{y^3}^1f(x,y)\,dy\,dx$
%% WRONG
$\int_0^8\int_{x^3}^2f(x,y)\,dy\,dx$
%% WRONG
$\int_0^8\int_0^{x^{1/3}}f(x,y)\,dy\,dx$


%15
%% PROBLEM
The area of the region bounded by $y = 9 - x^2$ and the $x$-axis is
36.  Find the centroid of this region.

\centerline{\epsfxsize=1.5truein\epsfbox{exam2.2.eps}}

%% CORRECT
$(0,3.6)$
%% WRONG
$(0,4.0)$
%% WRONG
$(0,2.4)$
%% WRONG
$(0,3.2)$
%% WRONG
$(0,2.8)$



%% END
\bye
