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

\input epsf
\def\r{{\bf r}}
\def\v{{\bf v}}
\def\a{{\bf a}}
\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{\vskip0.5in}

%% TITLE
\noindent
{\bf MATH 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\quad{\sl September 25, 2001}\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
{\it you must show your work and all important steps to receive credit}.

\medskip
You may use a calculator if you wish.

\vskip0.5truein

\def\answerskip{\vskip0.25in}


%1
%% PROBLEM
Find the angle between the planes $x - y + 2z = 3$ and $2x + y + z = -5$.

%% CORRECT
$\pi/3$
%% WRONG
$\pi/6$
%% WRONG
$\pi/2$
%% WRONG
$3\pi/4$
%% WRONG
$\pi/4$

%2
%% PROBLEM
Compute the arc length of the curve parameterized by $\r(t) = 2t \i +
t^{2} \j +  {1\over3} t^{3} \k$ for $0 \leq t \leq 1$.

%% CORRECT
${7\over3}$
%% WRONG
${\sqrt{46}\over3}$
%% WRONG
${17\over12}$
%% WRONG
${10\over3}$
%% WRONG
${\sqrt{161}\over12}$

%3
%% PROBLEM
Find the parametric equations of the line tangent to the curve
$\r(t) = \langle t^{2}, t,  \ln t \rangle$ at the point $(1,1,0)$.

%% CORRECT
$x=1+2t$, $y=1+t$, $z=t$
%% WRONG
$x=2t$, $y=1$, $z={1\over t}$
%% WRONG
$x=2+t$, $y=1+t$, $z=1$
%% WRONG
$x=1+2t^{2}$, $y=1+t$, $z=1$
%% WRONG
$x=1+t$, $y=1+2t$, $z=-t$

%4
%% PROBLEM
The position of a moving particle at time $t$ is given by
$\r(t) = \langle \cos(2t), t, \sin(2t) \rangle$. Determine the
tangential and normal components of the acceleration at $t=0$.

%% CORRECT
$a_{T} = 0$, $a_{N} = 4$
%% WRONG
$a_{T} = -4$, $a_{N} = 0$
%% WRONG
$a_{T} = \sqrt{5}$, $a_{N} = \sqrt{11}$
%% WRONG
$a_{T} = \sqrt{5}$, $a_{N} = 0$
%% WRONG
$a_{T} = 1$, $a_{N} = \sqrt{5}$

%5
%% PROBLEM
Determine which of the following plots represents the curve traced by
the vector function $\r(t) = \langle 2\cos(4t), 2\sin(4t), t \rangle$.

%% CORRECT
\epsfysize=2truein\epsfbox{graph1.eps}
%% WRONG
\epsfysize=2truein\epsfbox{graph2.eps}
%% WRONG
\epsfysize=2truein\epsfbox{graph3.eps}
%% WRONG
\epsfysize=2truein\epsfbox{graph4.eps}
%% WRONG
\epsfysize=2truein\epsfbox{graph5.eps}

%6
%% PROBLEM
Determine which of the following symmetric equations gives a line that
is parallel to the vector $\i + 2\j + (1/2)\k$.

%% CORRECT
$2(x+1) = y - 3 = 4(z-1)$
%% WRONG
$x + 1 = 2y + 1 = (1/2)z + 1$
%% WRONG
$2x = y = (1/2)z$
%% WRONG
$x=2y$ and $z=1$
%% WRONG
$y = (1/4)z$ and $x=0$

%7
%% PROBLEM
Find a vector that is orthogonal to the plane containing the
points $(1, 0, 1)$, $(2, -1, 1)$, and $(0, 0, 3)$.

%% CORRECT
$2\i + 2\j + \k$
%% WRONG
$-\i + \k$
%% WRONG
$2\i - \j - 2\k$
%% WRONG
$2\i + 3\j + 4\k$
%% WRONG
$\i  + \j - \k$

%8
%% PROBLEM
Determine the function that corresponds to the following contour map.
$$\epsfysize=1.5truein\epsfbox{contours.eps}$$

%% CORRECT
$f(x,y) = y e^x$
%% WRONG
$f(x,y) = xy - 1$
%% WRONG
$f(x,y) = x^2 - y$
%% WRONG
$f(x,y) = y e^{-x}$
%% WRONG
$f(x,y) = x - y^2$

%% END

\vfill\eject

\noindent\llap{9.\ }Find a vector function that traces out the curve
of intersection of the ellipsoid $(x-1)^2 + 3(y-1)^2 + (z-1)^2 = 1$
and the plane $y=x$.

\vfill\eject

\noindent\llap{10.\ }The force acting on a moving particle of mass $m=2$
at time $t$ is
  $$\F(t) = e^t\i + e^{-t}\j + t\k$$
If the particle's initial position and velocity are $\r(0) = \langle
0, 0, 1 \rangle$ and $\v(0) = \langle 0, 1, 0 \rangle$, respectively,
determine the particle's position vector, $\r(t)$, for any $t\ge 0$.

\vfill\eject

\noindent\llap{11.\ }Let $\r(t) = t\i + t\j + {t^2\over2}\k$. Find the
unit tangent vector, $\T(0)$, the unit normal vector, $\N(0)$, and the
unit binormal vector $\B(0)$ at $t=0$.

\vfill\eject

\noindent\llap{12.\  }Evaluate the limit
  $$\lim_{(x,y)\to(0,0)} {{y^4 + 2x^3y}\over{(x^2+y^2)^2}}$$
or show it does not exist.

\bye
