%\magnification=\magstep1

\input epsf
\def\r{{\bf r}}
\def\v{{\bf v}}
\def\a{{\bf a}}
\def\F{{\bf F}}
\def\T{{\bf T}}
\def\N{{\bf N}}
\def\B{{\bf B}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}

\def\frac#1#2{{{#1}\over{#2}}}
\def\iint{\int\!\!\!\int}
\def\iiint{\int\!\!\!\int\!\!\!\int}

\def\newline{\hfil\break}
\def\problemskip{\vskip0.5in}

%% TITLE
\noindent
{\bf MATH 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\quad{\sl November 12, 2002}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}

\vskip0.5truein
\noindent
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this page. There are 8 multiple
choice questions worth 5 points each and 4 partial credits problems worth
10 points each. You start with 20 points.  On the partial credit problems
{\it you must show your work and all important steps to receive credit}.

\medskip
You may use a calculator if you wish.

\vskip0.75truein

\def\answerskip{\vskip0.25in}


%1
%% PROBLEM
$x^2 +2y^2 +z^2 -xz=6$ at the point $(0,1,2)$.

%% CORRECT
$x-2y-2z=-6$
%% WRONG
$x-y-z=-3$
%% WRONG
$x-2y=-2$
%% WRONG
$2x-y-z=-3$
%% WRONG
$x+2y+2z =0$

%2
%% PROBLEM
Determine which of the following statements is true about the
function\hfil\break $f(x,y) = xy + x^4/4 + y^3/3$.

%% CORRECT
$(0,0)$ is a saddle point and $(-1,1)$ is a local minimum.
%% WRONG
$(0,0)$ is a local maximum and $(-1,1)$ is a local minimum.
%% WRONG
$(0,0)$ is a saddle point and $(-1,1)$ is a local maximum.
%% WRONG
$(0,0)$ and $(-1,1)$ are both saddle points.
%% WRONG
$(0,0)$ and $(-1,1)$ are both local maxima.


%3
%% PROBLEM
Evaluate $\int_0^1 \int_x^1 e^{y^2} dy\,dx$ by first reversing the order of
integration.

%% CORRECT
$(e-1)/2$
%% WRONG
$e-1$
%% WRONG
$e/2$
%% WRONG
$e$
%% WRONG
$2-e/2$

%4
%% PROBLEM
Let $E$ be the solid region below the inverted cone $z = 1 -
\sqrt{x^2+y^2}$ and above the $xy$-plane. If the density is uniform,
$\rho(x,y,z) = 1$, and the total mass of $E$ is $\pi/3$, find the
$z$-coordinate of the center of mass of $E$.

%% CORRECT
1/4
%% WRONG
1/3
%% WRONG
1/2
%% WRONG
3/8
%% WRONG
2/9

%5
%% PROBLEM
Let $D$ be the region in the plane bounded by the curves $y=x$ and
$y=x^2$. Evaluate $\int \int_D x^2 dA$.

%% CORRECT
$1/20$
%% WRONG
$1/12$
%% WRONG
$1/6$
%% WRONG
$1/28$
%% WRONG
$1/14$

%6
%% PROBLEM
Find the area of the part of the surface $z=1+x$ lying above the triangle
in the $xy$-plane with vertices $(0,0)$, $(1,0)$ and $(0,1)$.

%% CORRECT
$\sqrt{2}/2$
%% WRONG
$\sqrt{2}$
%% WRONG
$\sqrt{3}$
%% WRONG
$\sqrt{3}/3$
%% WRONG
$1/4$


%7
%% PROBLEM
Let $E$ be the solid region in the first octant bounded by the planes
$x=0$, $y=0$, $z=0$, $x+z = 2$, and $y+z = 2$, and lying below the plane
$z = 1$. Find $\iiint_E y\,dV$.
$$\epsfxsize=1.5in\epsfbox{exam2.7.eps}$$

%% CORRECT
$15/8$
%% WRONG
$7/4$
%% WRONG
$11/6$
%% WRONG
$2$
%% WRONG
$3/2$

%8
%% PROBLEM
Determine which of the following integrals gives the mass of the solid
region between the spheres $x^2 + y^2 + z^2 = 4$ and $x^2 + y^2 + z^2 =
2z$ above the $xy$-plane, if the density is given by the distance to
the $z$-axis.
$$\epsfxsize=1.5in\epsfbox{exam2.8.eps}$$

%% CORRECT
$\int_0^{2\pi}\int_0^{\pi/2}\int_{2\cos\phi}^2
\rho^3\sin^2\phi\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^{\pi}\int_{0}^{2\cos\phi}
\rho^3\sin\phi\,\cos\phi\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^{\pi/2}\int_{2}^{2\sin\phi}
\rho^3\sin^2\phi\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^{\pi}\int_{2\sin\phi}^4
\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^{\pi/2}\int_{0}^4
\rho^3\sin^2\phi\,\cos\phi\,d\rho\,d\phi\,d\theta$


%% END

\vfill\eject

\noindent\llap{9.~}%
Find the absolute maximum and minimum values of the function $f(x,y)=
4x-4x^2+y^2$ on the closed triangular region with vertices $(0,0),
(1,0),$ and $(0,1)$.


\vfill\eject

\noindent\llap{10.~}%
Use Lagrange multipliers to find the absolute maximum and minimum
of\newline $f(x,y,z) = 4x - 2z$ subject to the constraint $x^2 + y^2 +
z^2 = 16$.

\vfill\eject

\noindent\llap{11.~}%
Find the area of the region {\it inside} the cardioid $r=1+\sin\theta$
and {\it outside} the cardioid $r=1-\sin\theta$.
$$\epsfxsize=1.5in\epsfbox{exam2.11.eps}$$


\vfill\eject

\noindent\llap{12.~}%
Let $R$ be the region inside the square with vertices $(\pm1,0)$
and $(0,\pm1)$.  Evaluate\newline
  $\iint_R \sqrt{1 + y - x}\,\cos(y+x)\,dA$
by making an appropriate change of variables.\newline
[Hint: Start with $u = y+x$ and $v = y-x$.]
$$\epsfxsize=1.5in\epsfbox{exam2.12.eps}$$

\bye
