\documentclass{article}

\usepackage{graphics}

\renewcommand{\d}{\partial}
\renewcommand{\a}{\mathbf{a}}
\renewcommand{\i}{\mathbf{i}}
\renewcommand{\j}{\mathbf{j}}
\renewcommand{\k}{\mathbf{k}}
\newcommand{\n}{\mathbf{n}}
\renewcommand{\r}{\mathbf{r}}
\renewcommand{\v}{\mathbf{v}}
\newcommand{\w}{\mathbf{w}}
\renewcommand{\O}{\mathbf{0}}
\newcommand{\F}{\mathbf{F}}
\newcommand{\CC}{\mathcal{C}}
\newcommand{\curl}{\mathop{\mathrm{curl}}}
\newcommand{\iint}{\int\!\!\!\int}
\newcommand{\iiint}{\int\!\!\!\int\!\!\!\int}
\newcommand{\comp}{\mathrm{comp}}

\setlength{\textheight}{9in}
\setlength{\voffset}{-.75in}
\setlength{\textwidth}{6.5in}
\setlength{\hoffset}{-.75in}

\def\problemspace{0.5in}

\begin{document}


%% TITLE

\noindent
\textbf{MATH 225: Calculus III}\hfill
Name:\makebox[2.5in][l]{\rule{2in}{.5pt}}

\medskip\noindent
\textbf{Final Exam}\quad{\sl December 17, 2003}\hfill
Instructor:\makebox[2.5in][l]{\rule{2in}{.5pt}}

\vspace{.5in}
\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\noindent
You may use a calculator if you wish.

\vspace{.5in}

\def\answerspace{25pt}


%% PROBLEM 1

Find the scalar projection, $\comp_\w(\v)$, of the vector $\v=\langle 1, 1,
2\rangle$ onto the vector $\w = \langle 2, -2, 1\rangle$.
\[ \resizebox{1.5in}{!}{\includegraphics{final.1.eps}} \]

%% CORRECT

$2/3$

%% WRONG

$2/\sqrt{6}$

%% WRONG

$2$

%% WRONG

$\sqrt{6}/9$

%% WRONG

$1$


%% PROBLEM 2

Calculate the arc length of the helix parameterized by $\r(t)=\big\langle
4t, 3\cos(t), 3\sin(t)\big\rangle$ for $0\le t\le 2\pi$.

%% CORRECT

$10\pi$

%% WRONG

$0$

%% WRONG

$2\pi$

%% WRONG

$6\pi$

%% WRONG

$12\pi$


%% PROBLEM 3

Find a direction vector for the line of intersection of the planes
$x+y+2z=1$ and $3x-y=0$.
                                                                                
%% CORRECT
                                                                                
$\langle  1,3,-2\rangle$
                                                                                
%% WRONG
                                                                                
$\langle 3,1,-2\rangle$
                                                                                
%% WRONG
                                                                                
$\langle 3,-2,1\rangle$
                                                                                
%% WRONG
                                                                                
$\langle -2,3,1\rangle$
                                                                                
%% WRONG
                                                                                
$\langle -2,1,3\rangle$


%% PROBLEM 4

A particle starts from rest at time $t=0$ at the origin $(0,0,0)$. It then
begins to move with acceleration $\a(t)=\big\langle 1,t,t^2\big\rangle$.
Find the time, if ever, at which the particle passes through the point
$\big(2,\frac{4}{3},\frac{8}{3}\big)$.

%% CORRECT

never

%% WRONG

$t=1$

%% WRONG

$t=2$

%% WRONG

$t=3$

%% WRONG

$t=4$


%% PROBLEM  5

If $z = f(x,y)$, $x = u^{2}+v^{2}$ and $y=u^{2}-v^{2}$, find
$\frac{\partial^{2} z}{\partial u \partial v}$.

%% CORRECT

$4uv\big(f_{xx}-f_{yy}\big)$

%% WRONG

$4uv\big(f_{xx}+f_{yy}\big)$

%% WRONG

$4uv\big(f_{xx}+2f_{xy}-f_{yy}\big)$

%% WRONG

$2u\big(f_{xx}+2f_{xy}+f_{yy}\big)$

%% WRONG

$2\big(f_{xx}+f_{yy}\big)$


%% PROBLEM  6

Find the maximum rate of change of $f(x,y) = x^{2}y+2y$ at the point
$(-1,2)$ and the direction in which it occurs.

%% CORRECT

The maximum rate of change is $5$ in the direction of $-4\i + 3\j$.

%% WRONG

The maximum rate of change is $6$ in the direction of $4\i - 3\j$.

%% WRONG

The maximum rate of change is $-5$ in the direction of $-4\i + 3\j$.

%% WRONG

The maximum rate of change is $4$ in the direction of $-3\i + 4\j$.

%% WRONG

The maximum rate of change is $-4$ in the direction of $3\i - 4\j$.


%% PROBLEM  7

Find the minimum of the distance to the origin of the points on the surface
$xy^{2}z^{2}=4$.

%% CORRECT

$\sqrt{5}$ %$(1, \pm \sqrt{2}, \pm \sqrt{2})$

%% WRONG

$5$  %$(\pm 1, \pm \sqrt{2}, \pm \sqrt{2})$

%% WRONG

$1$ %$(2,\pm \sqrt{2} ,\pm \sqrt{2})$

%% WRONG

$\sqrt{3}$ %$(2,\pm 2, \pm 2)$

%% WRONG

$\sqrt{7}$ %$(4,0,0)$


%% PROBLEM  8

Find the maximum and minimum values of $f(x,y) = 2x^{2}+4x+y^{2}-2$ on the
disk $x^{2}+y^{2}\le 4$.

%% CORRECT

maximum is 14, minimum is $-4$

%% WRONG

maximum is 16, minimum is 0

%% WRONG

maximum is 20, minimum is $-4$

%% WRONG

maximum is 0, minimum is $-4$

%% WRONG

maximum is 16, minimum is $-6$


%% PROBLEM 9

Compute $\iiint_E \big(\sqrt{x^2 + y^2} + z\big)\, dV$ where $E$ is the
lower half of the solid sphere $x^2 + y^2 + z^2 \le 9$, $z \le 0$.

%% CORRECT

$\frac{81\pi}{2}\big(\frac{\pi}{4} -\frac{1}{2}\big)$

%% WRONG

$\frac{81\pi}{2}\big(\frac{\pi}{4} +\frac{1}{2}\big)$

%% WRONG

$\frac{81\pi}{2}\big(\frac{\pi}{2}-\frac{1}{2}\big)$

%% WRONG

$\frac{81\pi}{4}\big(\frac{\pi}{4} -\frac{1}{2}\big)$

%% WRONG

$\frac{108\pi}{3}$


%% PROBLEM 10

Determine which of the following integrals gives the area of the region in
the $xy$-plane below the $x$-axis, above $y=x^2 - 2$, and to the left of
$y=-2x-2$.

%% CORRECT

$\int_{-2}^0\int_{-\sqrt{y+2}}^{-y/2-1}\,dx\,dy$

%% WRONG

$\int_{-2}^0\int_{\sqrt{y+2}}^{-y/2-1}\,dx\,dy$

%% WRONG

$\int_{-\sqrt{2}}^0\int_{x^2-2}^{0}\,dy\,dx$

%% WRONG

$\int_{-\sqrt{2}}^0\int_{x^2-2}^{-2x-2}\,dy\,dx$

%% WRONG

$\int_{-2}^0\int_{-\sqrt{y+2}}^{1-y/2}\,dx\,dy$


%% PROBLEM 11

Determine which of the following integrals gives the volume of the region
bounded by the cylinder $x^2 + y^2 = 1$, and the planes $z=0$ and $y+z=1$.
\[ \resizebox{.5in}{!}{\includegraphics{final.11.eps}} \]

%% CORRECT

$ \int_{0}^{2\pi} \int_{0}^{1}\int_{0}^{1-r\sin(\theta)} r\,dz\,dr\,d\theta$

%% WRONG

$ \int_{0}^{2\pi} \int_{0}^{1}\int_{0}^{1-r\sin(\theta)} \,dz\,dr\,d\theta$

%% WRONG

$ \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}}\int_{0}^{1-y} dz\,dy\,dx$

%% WRONG

$ \int_{0}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int_{0}^{1-y} dz\,dy\,dx$

%% WRONG

$ \int_{0}^{\pi} \int_{0}^{1}\int_{0}^{1-r\sin(\theta)} r\,dz\,dr\,d\theta$


%% PROBLEM  12

Express the area between the ellipses $x^2 + \frac{y^2}{4} = 1$ and $x^2 +
\frac{y^2}{4} = 4$ as an integral, using the substitution $x=r\cos(\theta)$
and $y=2r\sin(\theta)$.
\[ \resizebox{.5in}{!}{\includegraphics{final.12.eps}} \]

%% CORRECT

$\int_{0}^{2\pi} \int_1^2 2r dr\,d\theta $

%% WRONG

$\int_{0}^{2\pi} \int_1^2 r dr\,d\theta $

%% WRONG

$\int_{0}^{2\pi} \int_1^4 2r dr\,d\theta $

%% WRONG

$\int_{0}^{2\pi} \int_1^4 r dr\,d\theta $

%% WRONG

$\int_{0}^{2\pi} \int_1^2  dr\,d\theta $


%% PROBLEM  13

Compute the line integral $\int_\CC x\,ds$ where $\CC$ is the curve
parameterized by $\r(t) = t^2\i + t^4\j$, $0 \le t \le 1$.

%% CORRECT

$\big(5^{3/2}-1\big)/12$

%% WRONG

$\big(\sqrt{3}-1\big)/6$

%% WRONG

$\big(\sqrt{6}-1\big)/2$

%% WRONG

$\big(3^{5/2}-1\big)/4$

%% WRONG

$\big(2^{3/2}-1\big)/3$


%% PROBLEM  14

Let $\CC$ be the curve $\r(t) = \big\langle t, \cos(2t), 1+\sin(3t)\big\rangle$,
$0\le t \le \pi/2$, and let
  \[ \F(x,y,z) = \big\langle y(2x+z), x(x+z)-z, y(x-1)+2z \big\rangle \]
Compute $\int_\CC \F\cdot d\r$.

%% CORRECT

$-\pi^2/4$

%% WRONG

$\pi^2/2-1$

%% WRONG

$1-\pi/4$

%% WRONG

$\pi/2$

%% WRONG

$0$


%% PROBLEM  15

Let $\CC$ be the triangle with vertices $(0,0)$, $(1,1)$, and $(2,0)$
oriented counterclockwise. Compute
  \[ \int_\CC \big(\cos(x^4) + x^2y^3\big)\,dx +
  \big(\sin(y^4) + x^3y^2\big)\,dy \]

%% CORRECT

$0$

%% WRONG

$2/5$

%% WRONG

$17/30$

%% WRONG

$-1$

%% WRONG

$1$


%% PROBLEM  16

Determine two vectors that are tangent to the surface
$\r(u,v) = \langle vu^2-2u, uv^2 - v, uv\rangle$ at the point $(0,1,2)$.

%% CORRECT

$\langle2,1,1\rangle$, $\langle4,3,2\rangle$

%% WRONG

$\langle0,1,1\rangle$, $\langle4,1,1\rangle$

%% WRONG

$\langle2,4,2\rangle$, $\langle1,3,1\rangle$

%% WRONG

$\langle0,2,-1\rangle$, $\langle1,-6,3\rangle$

%% WRONG

$\langle1,2,-1\rangle$, $\langle1,-2,1\rangle$


%% PROBLEM  17

Determine which of the following integrals gives the area of the surface
parameterized by $\r(u,v) = \big\langle u \cos(v), u^2/2, u
\sin(v)\big\rangle$, $0 \le u \le 1$, $0 \le v \le 2\pi$.

%% CORRECT

$\int_0^{2\pi}\int_0^1u\sqrt{1+u^2}\,du\,dv$

%% WRONG

$\int_0^{2\pi}\int_0^1 \sqrt{u^2(\cos(v)+\sin(v))+u(\sin^2(v)-\cos^2(v))}
\,du\,dv$

%% WRONG

$\int_0^{2\pi}\int_0^1\sqrt{1+u^2}\,du\,dv$

%% WRONG

$\int_0^{2\pi}\int_0^1\frac{u}{2}\sqrt{4+u^2}\,du\,dv$

%% WRONG

$\int_0^{2\pi}\int_0^1 \sqrt{u(\cos(v)+\sin(v)) +u^2/2} \,du\,dv$


%% PROBLEM  18

Let $S$ be the part of the cylinder $x^2 + z^2 = 1$, with $z\ge 0$, and
$0\le y\le 1$, and let $S$ have the upward orientation.  Determine which of
the following equals $\iint_S \F\cdot\n\,dS$ where $\F(x,y,z)=e^z\k$.
\[ \resizebox{1in}{!}{\includegraphics{final.18.eps}} \]

%% CORRECT

$\int_0^1\int_{0}^{\pi} e^{\sin(\theta)}\sin(\theta)\,d\theta\,dy$

%% WRONG

$\int_0^1\int_{0}^{\pi} -e^{\sin(\theta)}\cos(\theta)\,d\theta\,dy$

%% WRONG

$\int_0^1\int_{0}^{2\pi} e^{\sin(\theta)}\cos(\theta)\,d\theta\,dy$

%% WRONG

$\int_0^1\int_{-\pi/2}^{\pi/2}
e^{\sin(\theta)}\big(\sin(\theta)-\cos(\theta)\big)\,d\theta\,dy$

%% WRONG

$\int_0^1\int_{0}^{\pi/2} e^{\sin(\theta)}\sin(\theta)\,d\theta\,dy$


%% PROBLEM  19

Let $\CC$ be the intersection of the cylinder $x^2+y^2=1$ and the plane
$z=2y+3$ oriented counterclockwise when viewed from above. Calculate
$\int_\CC \F\cdot d\r$ where $\F(x,y,z)=\big\langle
2e^y-z,\cos(yz),xe^y\big\rangle$.
\[ \resizebox{1in}{!}{\includegraphics{final.19.1.eps}}\qquad
   \resizebox{.75in}{!}{\includegraphics{final.19.2.eps}} \]

%% CORRECT

$2\pi$

%% WRONG

$0$

%% WRONG

$\pi\sqrt{5}$

%% WRONG

$\pi$

%% WRONG

$\frac{\pi}{\sqrt{5}}$


%% PROBLEM  20

Find $\iint_{S} \F\cdot\n\,dS$ where $\F(x,y,z) = \big\langle xy, \frac{3}{4}y,
-zy\big\rangle$ and $S$ is the surface of the sphere $x^{2}+y^{2}+z^{2}=4$ with
the outward orientation.

%% CORRECT

$8\pi$

%% WRONG

$\frac{32}{3}\pi$

%% WRONG

$16\pi$

%% WRONG

$2\pi$

%% WRONG

$\pi$


%% END

\end{document}
