\magnification=\magstep1
\vsize=7.5in

\input epsf
\def\C{{\cal C}}
\def\r{{\bf r}}
\def\iint{\int\!\!\!\int}
\def\iiint{\int\!\!\!\int\!\!\!\int}

\def\problemskip{\vskip0.5in}

%% TITLE
\noindent
{\bf MATH 225: Calculus III}\hfill
{\bf Practice Exam II}\quad{\sl November 2002}

\def\problemskip{\vskip0.25in}

%1
%% PROBLEM
Suppose the plane $2x - y + z = 0$ is tangent to the level surface $f(x,y,z)
= 2$ at the point $(0,1,1)$. Determine which of the vectors below is
parallel to $\nabla f(0,1,1)$.

%% CORRECT
$\langle-2,1,-1\rangle$
%% WRONG
$\langle1,2,-1\rangle$
%% WRONG
$\langle2,-1,-1\rangle$
%% WRONG
$\langle4,-2,-1\rangle$
%% WRONG
$\langle4,2,-2\rangle$

%2
%% PROBLEM
Let $f(x, y) = 8 x^{3} + 6xy + y^{3}$. Determine which of the following
statements is true.

%% CORRECT
$(0, 0)$ is a saddle point and $(-1/2, -1)$ is a local maximum.
%% WRONG
$(0, 0)$ is a saddle point and $(-1/2, -1)$ is a local minimum.
%% WRONG
$(0, 0)$ is a local maximum and $(-1/2, -1)$ is a local minimum.
%% WRONG
$(0, 0)$ is a local minimum and $(-1/2, -1)$ is a local maximum.
%% WRONG
$(0, 0)$ is a local maximum and $(-1/2, -1)$ is a saddle point.

%3
%% PROBLEM
Let $C$ be the curve of intersection of the surfaces $x^{2}+y^{2}+z^{2}=1$
and $xy+z^{2} = 0$.  Determine which of the following systems of equations
must be solved to find the point(s) on $C$ closest to $(1, 1, 1)$ using
Lagrange multipliers.

%% CORRECT
  {\hsize=1.5truein\vtop{$$\displaylines{
  2(x-1) = 2\lambda x + \mu y \cr
  2(y-1) = 2\lambda y + \mu x \cr
   (z-1) =  \lambda z + \mu z \cr
  x^2 + y^2 + z^2 = 1 \cr
  xy + z^2 = 0
  }$$}}
%% WRONG
  {\hsize=1.5truein\vtop{$$\displaylines{
  2x =2\lambda (x-1) + \mu y \cr
  2y =2\lambda (y-1) + \mu x \cr
  2z = \lambda (z-1) + \mu z \cr
  x^2 + y^2 + z^2 = 1 \cr
  xy+z^{2} = 0
  }$$}}
%% WRONG
  {\hsize=1.5truein\vtop{$$\displaylines{
  (x-1) = \lambda x  \cr
  (y-1) = \lambda y  \cr
  (z-1) = \lambda z  \cr
  x^2 + y^2 - xy = 1
  }$$}}
%% WRONG
  {\hsize=1.5truein\vtop{$$\displaylines{
  2(x-1) = \lambda y \cr
  2(y-1) = \lambda x \cr
   (z-1) = \lambda z  \cr
  x^2 + y^2 - xy = 1
  }$$}}
%% WRONG
  {\hsize=1.5truein\vtop{$$\displaylines{
  2(x-1) = \lambda(2x - y) \cr
  2(y-1) = \lambda(2y - x) \cr
  2(z-1) = \lambda(x + y) \cr
  x^2 + y^2 - xy = 1
  }$$}}

%4
%% PROBLEM
Reverse the order of integration in the double integral
$\int_{-1}^1 \int_0^{\sqrt{1-x^2}} f(x,y)\ dy\ dx$.

%% CORRECT
$\int_0^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} f(x,y)\ dx\ dy$
%% WRONG
$\int_0^{\sqrt{1-x^2}} \int_{-1}^1 f(x,y)\ dx\ dy$
%% WRONG
$\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int_0^1  f(x,y)\ dx\ dy$
%% WRONG
$\int_{-1}^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} f(x,y)\ dx\ dy$
%% WRONG
$\int_{-1}^1 \int_0^{\sqrt{1-y^2}} f(x,y)\ dx\ dy$

%5
%% PROBLEM
Evaluate the double integral $\int_0^1 \int_0^{x^2} xe^y \ dy \ dx$.

%% CORRECT
${1\over2}e-1$
%% WRONG
${1\over2}(e-1)$
%% WRONG
${1\over2}e$
%% WRONG
$e - {1\over2}$
%% WRONG
$e-1$

%6
%% PROBLEM
Find the area inside the curve $r = 2 + \cos(\theta)$.
%% CORRECT
${9\pi\over2}$
%% WRONG
${3\pi\over2}$
%% WRONG
${6\pi}$
%% WRONG
${15\pi\over2}$
%% WRONG
$3\pi$

%7
%% PROBLEM
Determine which of the following double integrals gives the area of the
part of the surface $z = x + y^3$ that lies above the region bounded by
$x = y^2$, $y = 0$, and $x = 1$.

%% CORRECT
$\int_0^1\int_{y^2}^1\sqrt{2 + 9y^4}\,dx\,dy$
%% WRONG
$\int_0^1\int_0^{\sqrt{x}}\sqrt{2 + 3y^2}\,dy\,dx$
%% WRONG
$\int_0^1\int_{\sqrt{x}}^1\sqrt{2 + 3y^2}\,dy\,dx$
%% WRONG
$\int_0^1\int_0^{y^2}\sqrt{2 + 9y^4}\,dx\,dy$
%% WRONG
$\int_0^1\int_{y^2}^1\sqrt{1 + x^2 + 9y^4}\,dx\,dy$

%8
%% PROBLEM
Let $E$ be the tetrahedron with vertices $(0,0,0)$, $(1,0,0)$, $(0,2,0)$,
and $(0,0,2)$. Express the integral $\iiint_E f(x,y,z)\,dV$ as an iterated
integral.

%% CORRECT
$\int_0^1\int_0^{2-2x} \int_0^{2-2x-y} f(x,y,z)\,dz\,dy\,dx$
%% WRONG
$\int_0^1\int_0^{1-y} \int_0^{(2-y-z)/2} f(x,y,z)\,dx\,dz\,dy$
%% WRONG
$\int_0^1\int_0^2 \int_0^2 f(x,y,z)\,dz\,dy\,dx$
%% WRONG
$\int_0^2\int_0^{1-y} \int_0^{2-x-2y} f(x,y,z)\,dz\,dx\,dy$
%% WRONG
$\int_0^2\int_0^1 \int_0^{2-y} f(x,y,z)\,dz\,dx\,dy$

%9
%% PROBLEM
Let $E$ be the region  between the spheres $x^2 + y^2 + z^2 = z$ and
$x^2 + y^2 + z^2 = 2z$.  Which of the following represents $\iiint_E
(x^2 + y^2) \ dV$ in spherical coordinates.

%% CORRECT
$\int_{0}^{2\pi}\int_{0}^{\pi/2}
\int_{\cos(\phi)}^{2\cos(\phi)} \rho^4 \sin^3(\phi)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_{0}^{2\pi}\int_{0}^{\pi/2}
\int_{\cos(\phi)}^{2\cos(\phi)}\rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_{0}^{2\pi}\int_{0}^{\pi/2}\int_1^2
\rho^4\sin(\phi)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{\cos(\theta)}^{2\cos(\theta)}
\rho^4\sin^3(\theta)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_{0}^{2\pi}\int_{-\pi/2}^{\pi/2}\int_{\cos(\phi)}^{2\cos(\phi)}
\rho^4\sin^3(\phi)\,d\rho\,d\phi\,d\theta$

%10
%% PROBLEM
Let $\C$ be the curve parameterized by $\r(t) = \langle t, t^2,
t^2\rangle$, $0\le t \le 1$.  Evaluate the line integral $\int_{\C} x\,ds$.

%% CORRECT
13/12
%% WRONG
91/3
%% WRONG
0
%% WRONG
1/2
%% WRONG
$(5^{3/2}-1)/24$

%% END

\problemskip

\noindent\llap{11.\ }%
Find the absolute maximum and minimum values of $f(x, y) = x^{2}+y^{2}+x+1$
on the unit disk $x^{2}+y^{2} \le 1$.

\problemskip

\noindent\llap{12.\ }%
Let $D$ be the triangular region in the plane with vertices $(0,0)$,
$(2,0)$, and $(1,2)$. Compute $\iint_{D} x dA$.

\problemskip

\noindent\llap{13.\ }%
A lamina of uniform density occupies the region $D$ bounded by the parabola
$y = 1- x^2$ and the $x$-axis. Find its center of mass.

\problemskip

\noindent\llap{14.\ }
Let $R$ be the region bounded by the hyperbolas $xy = 1$ and $xy = 9$ and
the lines $y = x$ and $y=4x$ in the first quadrant. Use the change
of variables $x = uv$, $y = u/v$, $u\ge0$, $v\ge0$, to evaluate
the integral $\iint_R x^2y\,dA$.

\bye
