\input calc
\input epsf

\def\problemspace{.25in}

%% TITLE
\noindent
{\bf Math 225: Calculus III}\hfill Name:\hbox to 2.5truein{\hrulefill}
\medskip\noindent
{\bf Final Exam}\quad{\sl May 9, 1995}\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 25 multiple
choice questions worth 5 points each. You start with 25 points.

\bigskip

%1
%% PROBLEM
A vector $\a$ is perpendicular to a vector $\b$ if
%% CORRECT
$\a\bdot\b = 0$
%% WRONG
$\a\times\b = 0$
%% WRONG
$\a + \b = 0$
%% WRONG
$\a = c\b$, some scalar $c$
%% WRONG
$\a\bdot(\b\times\a) = 0$

%2
%% PROBLEM
Compute the projection of $\a = \i + 2\k$ onto $\b = \i + 2\j + \k$.
%% CORRECT
${1\over2}\i + \j + {1\over2}\k$
%% WRONG
$\i + 2\j + 1\k$
%% WRONG
$\i  + 2\k$
%% WRONG
${3\over5}\i  + {6\over5}\k$
%% WRONG
${1\over5}\i + {2\over5}\j + {1\over5}\k$

%3
%% PROBLEM
Determine the parametric equation of the line through the points $(-1,0,3)$ and
$(2,1,-1)$.
%% CORRECT
$\r(t) = (-1 + 3t)\i + t\j + (3-4t)\k$
%% WRONG
$\r(t) = (-1 + 2t)\i + t\j + (3-t)\k$
%% WRONG
$\r(t) = (2 - t)\i + \j + (-1+3t)\k$
%% WRONG
$\r(t) = 3t\i  - 4t\k$
%% WRONG
$\r(t) = (2 + 3t)\i + (1+t)\j + (-1+4t)\k$

%4
%% PROBLEM
Determine the equation of the plane perpendicular to the line
$x = 3 + t$, $y = 1 - 2t$, $z = -1 + t$ at the point $(3,1,-1)$.
%% CORRECT
$x - 2y + z = 0$
%% WRONG
$2x - y = 3$
%% WRONG
$4x - y = 11$
%% WRONG
$3x + y - z = 11$
%% WRONG
$3x - 2y + z = 6$

%5
%% PROBLEM
Find $\lim_{(x,y)\to(0,0)}{(x+y)^2\over x^2 + y^2}$.
%% CORRECT
{\sl Does not exist}
%% WRONG
$1$
%% WRONG
$1\over2$
%% WRONG
$0$
%% WRONG
$-1$

%6
%% PROBLEM
Suppose the motion of a particle is described by
$\r(t) = \cos(-\pi t)\i + \sin(\pi t)\j + {t}\k$.
Find the particle's speed at time $t = 1$.
%% CORRECT
$3.297$
%% WRONG
$4.500$
%% WRONG
$3.765$
%% WRONG
$4.021$
%% WRONG
$2.920$

%7
%% PROBLEM
Two particles are moving along the paths
$$\eqalign{\r_1(t) &= t\i + t^2\j + 2t\k\cr
           \r_2(t) &= t^3\i + j + (3t-1)\k\cr}$$
and collide at time $t=1$ at the point $(1,1,2)$.
Find the angle in radians of this collision.
$$\epsfxsize=1.5in\epsfbox{final1.eps}$$

%% CORRECT
$0.785$
%% WRONG
$1.571$
%% WRONG
$3.142$
%% WRONG
$1.047$
%% WRONG
$0.524$

%8
%% PROBLEM
A right circular cylinder with volume $V$ has a height, $h$, and a
radius, $r$, that depend on the pressure $x$ and temperature $y$
of the surrounding medium. If $r = 2$, $h = 5$, $\grad r = -2\i + 3\j$,
and $\grad h = -\i + 4\j$, when $(x,y) = (10,20)$, which of the
following gives $\grad V$.
%% CORRECT
$-44\pi\i+76\pi\j$
%% WRONG
$\0$
%% WRONG
$20\pi\i + 4\pi\j$
%% WRONG
$-28\pi\i + -4\pi$
%% WRONG
$40\pi\i + 48\pi$

%9
%% PROBLEM
Let $f(x,y,z) = x y^2 + x^2z + z^2y$. Find the derivative of $f$ in the
direction of the vector $\i + 2\j + 2\k$ at the point $(1,-1,2)$.
%% CORRECT
$1$
%% WRONG
$2$
%% WRONG
$3$
%% WRONG
$4$
%% WRONG
$6$

%% PROBLEM
Determine the equation of the plane tangent to the graph of
$f(x,y) = e^y(\sin(x) - \cos(x))$ at the point $(0,0,-1)$.
%% CORRECT
{$x - y - z = 1$}
%% WRONG
{$2e^y \sin(x) - z = 1$}
%% WRONG
{$e^y(\cos(x)x + \sin(x)) - e^y(\cos(x) - \sin(x))y = 0$}
%% WRONG
{$x + y + z = -1$}
%% WRONG
{$x - y = 0$}


%11
%% PROBLEM
Let $f(x,y) = x^3 - 3xy + y^2$. Determine which of the following statements is
true.
%% CORRECT
$f$ has a relative minimum at $({3\over2},{9\over4})$.
%% WRONG
$f$ has a relative maximum at $({3\over2},{9\over4})$.
%% WRONG
$f$ has a saddle point at $({3\over2},{9\over4})$.
%% WRONG
$({3\over2},{9\over4})$ is {\bf not} a critical point of $f$
%% WRONG
{\sl None of the above}

%12
%% PROBLEM
Find the maximum of $f(x,y) = x(1 - y)$ on the unit circle $x^2 + y^2 = 1$.
%% CORRECT
$1.299$
%% WRONG
$0.707$
%% WRONG
$0.250$
%% WRONG
$0.866$
%% WRONG
$1.414$

%13
%% PROBLEM
Reverse the order of integration in the iterated integral
$\int_{-1}^1\int_{2|x|}^2f(x,y)\dydx$.
%% CORRECT
$\int_0^2\int_{-y/2}^{y/2}f(x,y)\dxdy$
%% WRONG
$\int_{2|x|}^2\int_{-1}^{1}f(x,y)\dxdy$
%% WRONG
$\int_{-1}^1\int_{0}^{|y|/2}f(x,y)\dxdy$
%% WRONG
$\int_0^2\int_0^{y/2}f(x,y)\dxdy$
%% WRONG
$\int_{-2}^2\int_{-y/2}^{0}f(x,y)\dxdy$

%14
%% PROBLEM
Determine which of the following integrals gives the volume of the
solid bounded by the planes $x + y - z = 0$, $x - 2y + z = 0$, $y =
0$, and $x + y = 2$.

%% CORRECT
$\int_0^{4/3}\int_{y/2}^{2-y}\int_{2y-x}^{x+y}dz\,dx\,dy$
%% WRONG
$\int_0^{2}\int_0^{2-x}\int_{2y-x}^{x+y}dz\,dy\,dx$
%% WRONG
$\int_0^{2}\int_0^{2-y}\int_{2y-x}^{x+y}dz\,dx\,dy$
%% WRONG
$\int_0^{4/3}\int_{2x}^{2-x}\int_{x+y}^{2y-x}dz\,dy\,dx$
%% WRONG
$\int_0^{2}\int_y^{2-y}\int_{x+y}^{2y-x}dz\,dx\,dy$

%15
%% PROBLEM
Let $D$ be the solid in the first octant that is bounded above by the plane
$z = 9$ and is bounded below by $z = x^2 + y^2$. Write
$\iiint_D xyz\,dV$ in cylindrical coordinates.
%% CORRECT
$\int_0^{\pi/2}\int_0^3\int_{r^2}^9 zr^3\cos(\theta)\sin(\theta)\,dz\drdw$
%% WRONG
$\int_0^{\pi}\int_0^3\int_0^9 zr^2\cos(\theta)\sin(\theta)\,dz\drdw$
%% WRONG
$\int_0^{2\pi}\int_0^1\int_0^3 zr^2\cos(\theta)\sin(\theta)\,dz\drdw$
%% WRONG
$\int_0^{\pi/2}\int_0^3\int_r^9 zr^3\cos(\theta)\sin^2(\theta)\,dz\drdw$
%% WRONG
$\int_0^{\pi/2}\int_0^1\int_{r^2}^3 zr^2\cos(\theta)\sin(\theta)\,dz\drdw$

%16
%% PROBLEM
Compute $\iiint_D \sqrt{x^2 + y^2}\,dV$ where $D$ is the solid sphere $x^2 + y^2 +
z^2 \le 4$.
%% CORRECT
$4 \pi^2$
%% WRONG
$\pi^2$
%% WRONG
$12\pi/5$
%% WRONG
$16\pi^2/3$
%% WRONG
$16\pi/5$

%17
%% PROBLEM
Let $R$ be the region defined by
${1\over2}x\le y \le 2x$, $1 \le xy \le 2$.
Using the change of variables $u = xy$, $v = {y\over x}$
transform  $\iint_R xy \,dA$ into an iterated integral in the
$uv$-plane. (Hint: You do not have to solve for $x$ and $y$.)
$$\epsfxsize=3in\epsfbox{final2.eps}$$

%% CORRECT
{$\int_{1/2}^2\int_1^2{u\over2v}\,du \,dv$}
%% WRONG
{$\int_{1/2}^2\int_1^2 uv \,du\,dv$}
%% WRONG
{$\int_{1/2}^2\int_1^2 2uv^2 \,du \,dv$}
%% WRONG
{$\int_{x/2}^{2x}\int_1^2u\,du \,dv$}
%% WRONG
{$\int_1^2\int_{x/2}^{2x}\left(uv + {u\over v}\right)\,du\,dv$}

%18
%% PROBLEM
Compute $\int_{\CC} x\,ds$ where $\CC$ is the curve parameterized by
$\r(t) = \sqrt{t}\,\i + t\j$, $0\le t \le 1$.
%% CORRECT
{$(5\sqrt{5} - 1)/12$}
%% WRONG
{$2\sqrt{5} - 1$}
%% WRONG
{$(\sqrt{5} - 6)/2$}
%% WRONG
{$(36\sqrt{6} - 1)/8$}
%% WRONG
{$2\sqrt{6} - 3$}

%19
%% PROBLEM
Let $\CC$ be the curve $\r(t) = t\i + t^2\j + t\k$, $0\le t \le 1$. Compute $\int_\CC
x^3z\,dx + xy\,dy + 2 y z^2\,dz$.
%% CORRECT
$1$
%% WRONG
$1/5$
%% WRONG
$3/5$
%% WRONG
$4/5$
%% WRONG
$2/5$

%% PROBLEM
Let $\F = -z^2\i + e^z\j + (e^zy - 2xz)\k$ and let $\r(t) = t\i + t^2\j +
t^3\k$, $0 \le t \le 1$. Evaluate $\int_C \F\bdot d\r$.
%% CORRECT
$e-1$
%% WRONG
$1$
%% WRONG
$0$
%% WRONG
$e - e^{-1}$
%% WRONG
$2e-1$

%21
%% PROBLEM
Use Green's Theorem to compute $\int_{\cal C} (y^3 + y)\,dx + 3y^2x\,dy$
where $\cal C$ is $x^2 + y^2 = 9$ oriented counter-clockwise.
%% CORRECT
{$-9\pi$}
%% WRONG
{$3\pi$}
%% WRONG
{$0$}
%% WRONG
{$6\pi$}
%% WRONG
{$-9\pi/4$}

%22
%% PROBLEM
Let $\Sigma$ be the portion of the plane $2x + y - z = 0$ above the
rectangle $0 \le x \le 1$, $0 \le y \le 2$ in the $xy$-plane. Compute
$\iint_\Sigma z^2 - 4xy \, d\sigma$.
$$\epsfxsize=2in\epsfbox{final3.eps}$$

%% CORRECT
$16\sqrt{6}/3$
%% WRONG
$16/3$
%% WRONG
$32\sqrt{2}$
%% WRONG
$32\sqrt{3}$
%% WRONG
$32/3$

%23
%% PROBLEM
Let $\Sigma$ be the portion of the hyperboloid $z = x^2 - y^2$
inside the cylinder $x^2+y^2 = 1$ with an
upward pointing normal and let $\F(x,y,z) = 3x\i + 2y\j +
5z\k$. Compute $\iint_{\Sigma}\F\cdot\n\,d\sigma$.
$$\epsfxsize=2in\epsfbox{final4.eps}$$

%% CORRECT
{$-\pi/2$}
%% WRONG
{$2\pi$}
%% WRONG
{$-3\pi/2$}
%% WRONG
{$\pi$}
%% WRONG
{$-\pi$}

%24
%% PROBLEM
Let $\CC$ be the intersection of the paraboloid $x^2 + y^2 + z = 2$ and the
plane $x + y + z = 1$ oriented counter-clockwise when viewed from above.
Use Stokes' Theorem to compute
$\int_\CC x^2\,dx + y^2\,dy + z^2\,dz$.
$$\epsfxsize=2in\epsfbox{final5.eps}$$

%% CORRECT
$0$
%% WRONG
$\pi/3$
%% WRONG
$2\pi/3$
%% WRONG
$4\pi/3$
%% WRONG
$8\pi/3$

%25
%% PROBLEM
Let $\F = xe^{xy}\i - ye^{xy}\j + z\k$, let $\Sigma$ be the sphere
$x^2 + y^2 + z^2 = 4$, and let $\n$ be the outward normal vector to $\Sigma$.
Evaluate the flux integral $\iint_\Sigma\F\bdot\n\,d\sigma$.
%% CORRECT
$32\pi/3$
%% WRONG
$4\pi/3$
%% WRONG
$8\pi/3$
%% WRONG
$16\pi/3$
%% WRONG
$0$


%% END
\end
