\magnification=\magstep1
\vsize=7.5in
%\voffset=-.25in

\input epsf
\def\frac#1#2{{{#1}\over{#2}}}
\def\iint{\int\!\!\!\int}
\def\iiint{\int\!\!\!\int\!\!\!\int}
\def\curl{{\rm curl\,}}
\def\d#1/d#2{\frac{\partial #1}{\partial #2}}
\def\r{{\bf r}}
\def\v{{\bf v}}
\def\a{{\bf a}}
\def\b{{\bf b}}
\def\C{{\cal C}}
\def\F{{\bf F}}
\def\T{{\bf T}}
\def\N{{\bf N}}
\def\n{{\bf n}}
\def\B{{\bf B}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}


\def\problemskip{\vskip0.5in}

%% TITLE
\noindent
{\bf MATH 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\quad{\sl December 17, 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 20 multiple
choice questions worth 6 points each. You start with 30 points.

\medskip
You may use a calculator if you wish.

\vskip0.5truein

\def\answerskip{\vskip0.25in}


%1
%% PROBLEM
Let $\theta$ be the angle between the vectors $\a =
\langle-2,4,1\rangle$ and $\b =\langle-1,2,-1\rangle$.  Find $\cos
\theta$.

%% CORRECT
$\frac{3}{\sqrt{14}}$
%% WRONG
$\frac{9}{\sqrt{6}}$
%% WRONG
$\frac{9}{\sqrt{21}}$
%% WRONG
$\frac{1}{14}$
%% WRONG
$\frac{\sqrt{2}}{\sqrt{7}}$


%2
%% PROBLEM
The motion of a particle is described by $\r(t) = \langle 2t^3-2t, 3-2t^2,
3t- \ln t\rangle$, $t\ge 0$. Find the unit tangent vector at time $t=1$.

%% CORRECT
$\frac{1}{3}\langle2, -2, 1\rangle$
%% WRONG
$\frac{1}{\sqrt{5}}\langle0,1,2\rangle$
%% WRONG
$\frac{1}{3}\langle2, 2, 1\rangle$
%% WRONG
$\frac{1}{41}\langle4,4,3\rangle$
%% WRONG
$\frac{1}{\sqrt{5}}\langle-1,2,0\rangle$


%3
%% PROBLEM
Suppose $z$ is defined implicitly as a function of $x$ and $y$ by
$xyz=\ln(x+y+z)$. Find $\frac{\partial z}{\partial x}$.

%% CORRECT
$\frac{1-yz(x+y+z)}{xy(x+y+z)-1}$
%% WRONG
$\frac{yz}{1-xy(x+y+z)}$
%% WRONG
$\frac{1}{xy(x+y+z)}$
%% WRONG
$\frac{1+xy}{x+y+z+1}$
%% WRONG
$\frac{yz(x+y+z)}{xy(x+y+z)+1}$


%4
%% PROBLEM
Determine which of the following statements applies to the
function\hfill\break $f(x,y) = x^3 - 3x^2 + 3y^2$.

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


%5
%% PROBLEM
Find the maximum value of the function $f(x,y) = x - y$ on the ellipse
$4x^2 + y^2 = 1$.

%% CORRECT
$\frac{\sqrt{5}}{2}$
%% WRONG
$\frac{3\sqrt{3}}{4}$
%% WRONG
$\sqrt{2}$
%% WRONG
$\frac{1}{2}$
%% WRONG
$1$

%6
%% PROBLEM
Evaluate $\int_0^{1}\int_{x^2}^{1} x^5e^{y^4}\,dy\,dx$.

%% CORRECT
$\frac{e-1}{24}$
%% WRONG
$\frac{e}{4}-\frac{1}{4e}$
%% WRONG
$\frac{e}{12}$
%% WRONG
$\frac{e}{12}-1$
%% WRONG
$\frac{e-1}{4}$


%7
%% PROBLEM
Let $E$ be the solid tetrahedron with vertices $(0,0,0)$, $(1,0,0)$,
$(0,2,0)$, and $(0,0,4)$.\hfill\break Determine which of the following
integrals gives $\iiint_E f(x,y,z)\,dV$.

%% CORRECT
$\int_0^1\int_0^{2 - 2x}\int_0^{4 - 2y - 4x}f(x,y,z)\,dz\,dy\,dx$
%% WRONG
$\int_0^1\int_0^2\int_0^4f(x,y,z)\,dx\,dy\,dz$
%% WRONG
$\int_0^1\int_0^{2-x}\int_0^{x+2y}f(x,y,z)\,dz\,dy\,dx$
%% WRONG
$\int_0^2\int_0^{4-2y}\int_0^{4x+2y}f(x,y,z)\,dz\,dx\,dy$
%% WRONG
$\int_0^2\int_0^{1-2y}\int_0^{4-4x-2y}f(x,y,z)\,dz\,dx\,dy$

%8
%% PROBLEM
Find the average of the function $f(x,y,z) = z$ in the region under the
unit hemisphere sphere, $0 \le z \le \sqrt{1 - x^2 - y^2}$.

%% CORRECT
$3/8$
%% WRONG
$3/16$
%% WRONG
$1/2$
%% WRONG
$1/4$
%% WRONG
$5/16$

%9
%% PROBLEM
Let $f(x,y) = x^3 +  xy^2$ and suppose $x$ and $y$ are functions of
$u$ and $v$.  If $\d{x}/du = 3$,\hfill\break $\d{x}/dv = -2$, $\d{y}/du
= -4$ and $\d{y}/dv = 7$ when $(x,y) = (1,2)$, compute $\d{f}/du$ at
that point.

%% CORRECT
$5$
%% WRONG
$11$
%% WRONG
$-30$
%% WRONG
$14$
%% WRONG
$49$



%10
%% PROBLEM
Find the mass of a thin wire in the shape of the parabola $x = 1-y^2$,
$0\le y \le \sqrt{2}$, if the density is given by $\delta(x,y) = y$.

%% CORRECT
$13/6$
%% WRONG
$\frac{1}{3}(5^{3/2}-1)$
%% WRONG
$\frac{1}{12}(2^{3/4}-1)$
%% WRONG
$1$
%% WRONG
$11/12$

%11
%% PROBLEM
Let $\C$ be the curve $\r(t) = \langle t-1,t^2+1,\frac{1}{t}\rangle$, $1\le
t\le 2$.  Compute $\int_C (x^2-y+yz)dy$.

%% CORRECT
$-\frac{8}{3}$
%% WRONG
$\frac{4}{3}$
%% WRONG
$-\frac{3}{4}$
%% WRONG
$2$
%% WRONG
$0$

%12
%% PROBLEM
Find the equation of the plane tangent to the surface parameterized
by\hfill\break $\r(u,v) = \langle u^2, 2uv, v^3\rangle$ at the point
$(9,6,1)$.

%% CORRECT
$x - 3y + 6z = -3$
%% WRONG
$9x + 6y + z = 0$
%% WRONG
$6x + 2y + 3z = -69$
%% WRONG
$6x - 18y + z = -49$
%% WRONG
$x + y - 3z = 12$

%13
%% PROBLEM
Evaluate the surface integral $\iint_S z^2\,dS$ where $S$ is the part of
the cone $z = \sqrt{x^2 + y^2}$ between $z = 1$ and $z = 2$.

%% CORRECT
$15\pi\sqrt{2}/2$
%% WRONG
$8\pi\sqrt{3}$
%% WRONG
$13\pi\sqrt{3}/3$
%% WRONG
$7\pi\sqrt{2}/2$
%% WRONG
$14\pi/3$


%14
%% PROBLEM
Compute the flux integral $\iint_S \F\cdot\n\,dS$ where $\F(x,y,z) =
\langle y^2, -xy, z\rangle$ and $S$ is the paraboloid $z = x^2 + y^2$
below $z = 1$ with upward orientation.

%% CORRECT
$\pi/2$
%% WRONG
$\pi$
%% WRONG
$-\pi/3$
%% WRONG
$\pi/4$
%% WRONG
$-2\pi/3$

%15
%% PROBLEM
Calculate $\int_\C \F \cdot d\r$ where $\F(x,y,z)= \langle 2xz-y,
2y-x+z, x^2+y+2z\rangle$, and $\C$ is the curve $\r(t)=\langle t, t^2,
e^{t-t^2}\rangle$ for $0\le t \le 1$.

%% CORRECT
$2$
%% WRONG
$0$
%% WRONG
$3$
%% WRONG
$-1$
%% WRONG
$1$

%16
%% PROBLEM
Use Green's Theorem to calculate the area of the region in the plane
enclosed by the curve $\r(t)=\langle\cos t,\sin^3 t\rangle$, $0\le t
\le 2\pi$.
$$\epsfxsize=1.5in\epsfbox{final.7.eps}$$

%% CORRECT
$3\pi/4$
%% WRONG
$\pi/2$
%% WRONG
$\pi$
%% WRONG
$1$
%% WRONG
$2$

%17
%% PROBLEM
Find $\curl \F(0,0,0)$ where $\F(x,y,z)=\langle e^x \sin y,
e^x \cos y,x+z\rangle$.

%% CORRECT
$\langle0,-1,0\rangle$
%% WRONG
$\langle0,0,0\rangle$
%% WRONG
$\langle1,0,0\rangle$
%% WRONG
$\langle0,0,-1\rangle$
%% WRONG
$\langle1,1,1\rangle$

%18
%% PROBLEM
Let $\C$ be the intersection of the cylinder $(x-1/2)^2 + y^2 = 1/4$ and the
plane $x + y + z = 1$, oriented counterclockwise when viewed from above,
and let $\F(x,y,z) = \langle z - y, y, x\rangle$.
Calculate $\int_\C \F\cdot d\r$.
$$\epsfxsize=1.75in\epsfbox{final.18.1.eps}
\qquad\epsfxsize=1.75in\epsfbox{final.18.2.eps}$$

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

%19
%% PROBLEM
Let $S$ be the surface $z = 1 - x^2 - y^2$ above the $xy$-plane
with upward orientation, and let $\F(x,y,z)$ be a vector field
with $\curl\F(x,y,z) = \langle y-4xz,2yz-x,z^2 \rangle$. Compute
$\iint_S\curl\F\cdot\n\,dS$.

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

%20
%% PROBLEM
Let $S$ be the surface of the cube with vertices $(\pm1,\pm1,\pm1)$
with outward orientation.  Calculate the flux of the vector field
$\F(x,y,z) = \langle e^y, e^x, z\rangle$ across $S$.

%% CORRECT
$8$
%% WRONG
$4$
%% WRONG
$2$
%% WRONG
$0$
%% WRONG
$16$

%% END
\bye
