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

\input epsf
\def\frac#1#2{{{#1}\over{#2}}}
\def\newline{\hfil\break}
\def\d{\partial}
\def\comp{{\rm comp}}
\def\r{{\bf r}}
\def\v{{\bf v}}
\def\a{{\bf a}}
\def\b{{\bf b}}
\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\problemskip{\vskip0.75in}

%% TITLE
\noindent
{\bf MATH 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\quad{\sl October 1, 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
try to simplify your answer and indicate your final answer clearly.
{\it You must show your work and all important steps to receive credit}.

\medskip\noindent
You may use a calculator if you wish.

\vskip0.5truein

\def\answerskip{\vskip0.25in}


%1
%% PROBLEM
Consider the triangle with vertices $A = (0,1,1)$, $B = (-3,5,1)$,
and $C = (1,4,-1)$.  Find the angle in radians at $A$.

%% CORRECT
$\cos^{-1}\Big(\frac{9}{5\sqrt{14}}\Big)$
%% WRONG
$\cos^{-1}\Big(\frac{3\sqrt{2}}{\sqrt{35}}\Big)$
%% WRONG
$\cos^{-1}\Big(\frac{8\sqrt{2}}{3\sqrt{35}}\Big)$
%% WRONG
$\cos^{-1}\Big(\frac{16}{5\sqrt{21}}\Big)$
%% WRONG
$\cos^{-1}\Big(\frac{5}{7\sqrt{6}}\Big)$

%2
%% PROBLEM
Let $\a = \langle-1,4,8\rangle$ and $\b = \langle2,-3,6\rangle$.
Compute $\comp_{\a}\b$, the scalar projection of $\b$ onto $\a$.

%% CORRECT
34/9
%% WRONG
34/7
%% WRONG
34/63
%% WRONG
62
%% WRONG
$\sqrt{62}$

%3
%% PROBLEM
Compute the distance from the point $(-1,0,-2)$ to the plane $x-2y+3z=7$.

%% CORRECT
$\sqrt{14}$
%% WRONG
$\sqrt{14}/7$
%% WRONG
$\sqrt{68}/39$
%% WRONG
$\sqrt{68}$
%% WRONG
0


%4
%% PROBLEM
Find $\r(t)$ if $\r'(t) = \langle2t, -\sin(t), e^t\rangle$ and
$\r(0) = \langle2,-2,2\rangle$.

%% CORRECT
$\langle t^2+2, \cos(t)-3, e^t+1 \rangle$
%% WRONG
$\langle 2t^2, \cos(t)-3, e^t+1 \rangle$
%% WRONG
$\langle t^2+2, -\cos(t)-1, e^t+1 \rangle$
%% WRONG
$\langle 2t + 2, -\sin(t)-2, e^t+1 \rangle$
%% WRONG
$\langle 2t^2, \cos(t)-3, 2e^t \rangle$

%5
%% PROBLEM
Determine which of the following expressions gives the length of the
curve defined by $\r(t)= 2t\i + \cos t\j +2\sin t\k$ between the points
$(0,1,0)$ and $(2\pi,-1,0)$.

%% CORRECT
$\int_0^{\pi} \sqrt{4+\sin^2 t +4\cos^2 t}\,dt$
%% WRONG
$\int_0^{\pi} \sqrt{4t^2 +\cos^2 t +4\sin^2 t}\,dt$
%% WRONG
$\int_0^{2\pi} \sqrt{4t^2 +\cos^2 t +4\sin^2 t}\,dt$
%% WRONG
$\int_0^{2\pi} \sqrt{4+\sin^2 t +4\cos^2 t}\,dt$
%% WRONG
$\int_0^{\pi} (2\i -\sin t\j +2\cos t \k)\,dt$

%6
%% PROBLEM
Compute the limit $\lim_{(x,y)\to (0,0)} \frac{xy}{2x^2+y^2}$.

%% CORRECT
{\it does not exist}
%% WRONG
0
%% WRONG
$1/3$
%% WRONG
$1/2$
%% WRONG
1

%7
%% PROBLEM
Let $f(x,y) = [\sin(x^2) + \cos(y^2)]e^{-x^2}$. Compute
$f_{xy}(\sqrt{\pi/2},\sqrt{\pi/2})$.

%% CORRECT
$2\pi e^{-\pi/2}$
%% WRONG
$-4\pi e^{-\pi/2}$
%% WRONG
$-4\sqrt{\pi/2}e^{-\pi/2}$
%% WRONG
$\sqrt{\pi/2}e^{-\pi/2}$
%% WRONG
0

%8
%% PROBLEM
Suppose $f(x,y)$ satisfies $f_x(x,y) = -y/(x-y)^2$ and $f_y(x,y) =
x/(x-y)^2$. If $x = u^3 - v$ and $y = uv - v^2$, compute $\d f/\d u$
at the point $(u,v) = (1,-1)$.

%% CORRECT
$1/4$
%% WRONG
$-1/8$
%% WRONG
$1/2$
%% WRONG
$-1$
%% WRONG
$0$

%% END

\vfill\eject
%\bigskip

\noindent\llap{9.\ } Find a {\it unit} vector that is perpendicular to
both of the vectors $\a = \langle-1,2,1\rangle$ and\newline $\b =
\langle4,-1,3\rangle$.

\vfill\eject
%\bigskip

\noindent\llap{10.\ }
Find the point where the line defined by $\r(t) = \langle 2+t,t,3-3t
\rangle$ intersects the plane $3x - y + 2z = 4$.

\vfill\eject
%\bigskip

\noindent\llap{11.\ }
Let $C$ be the curve defined by $\r(t)= t \i + t^2 \j + t^3 \k$ and let
$M$ be the plane $x-3y+3z=0$.

\bigskip

(a) Find the points (if any) on $C$ where the {\it normal plane} to $C$
is parallel to~$M$.

\bigskip

(b) Find the points (if any) on $C$ where the {\it tangent line} to $C$
is parallel to~$M$.


\vfill\eject
%\bigskip

\noindent\llap{12.\ }Find the derivative of $f(x,y,z) = xy^2 - x^2 z + z^3$
at the point $(3,3,1)$ in the direction parallel to the line
$x = 1+2t$, $y = -t$, $z=3+2t$.

\bye
