\magnification=\magstep1
\input epsf
\input calc
\def\everymath{\displaystyle}
\def\problemspace{0.25truein}

%% TITLE
\noindent
{\bf Math 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\quad{\sl February  11, 1997}\hfill
Section:\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 15 multiple
choice questions worth 5 points each. You start with 25~points.

\def\answerspace{0.25in}


%% PROBLEM
Find the distance from the point $(2,1,-1)$ to the $z$-axis.
%% CORRECT
$\sqrt{5}$
%% WRONG
$1$
%% WRONG
$\sqrt{3}$
%% WRONG
$\sqrt{2}$
%% WRONG
$2$



%% PROBLEM
Find the area of the parallelogram with vertices at the points
$(0,0,0)$, $(-3,1,2)$, $(1,1,4)$, and $(-2,2,6)$.
%% CORRECT
$14.697$
%% WRONG
$13.250$
%% WRONG
$12.765$
%% WRONG
$15.913$
%% WRONG
$11.832$


%% PROBLEM
Which of the following vectors is perpendicular to  $\v = \i - 3\j + 5\k$.
%% CORRECT
$\i + 2\j + \k$
%% WRONG
$4\i + \j$
%% WRONG
$5\i + \k$
%% WRONG
$-\i + \j + \k$
%% WRONG
$2\i + 3\j - \k$


%% PROBLEM
Find the angle in radians between the vectors $\v = \i - 2\j + \k$ and
$\w = \i - \j$.
%% CORRECT
$\pi/6$
%% WRONG
$\pi/3$
%% WRONG
$\pi/4$
%% WRONG
$\pi$
%% WRONG
$4\pi/3$


%% PROBLEM
Let $\a = -\i + \j + 3\k$ and $\b = 4\i + \k$. Compute $\a \times \b$.
%% CORRECT
$\i + 13\j - 4\k$
%% WRONG
$-7$
%% WRONG
$-12\i + \j - 4\k$
%% WRONG
$-4\i - 3\k$
%% WRONG
$3\i + 5\j + 2\k$


%% PROBLEM
Find the point where the line $x = -5 + 2t$, $y = 6 - t$, $z = 2 + 5t$
intersects the plane $x - y + 2z = 6$.
%% CORRECT
$(-3,5,7)$
%% WRONG
$(-1,4,12)$
%% WRONG
$(-7,7,-3)$
%% WRONG
$(7,0,32)$
%% WRONG
$(0,0,2)$


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


%% PROBLEM
Calculate the distance from the origin to the plane $x + y - z = 1$.
%% CORRECT
$0.577$
%% WRONG
$1.0$
%% WRONG
$0.684$
%% WRONG
$0.967$
%% WRONG
$1.121$


%% PROBLEM
Find a vector perpendicular to the plane
$5x - 3y + z = 2$.
%% CORRECT
$5\i - 3\j + \k$
%% WRONG
$3\i - 5\j$
%% WRONG
$\i + 2\j - \k$
%% WRONG
$\i - \j - 2\k$
%% WRONG
$2\i + 3\j - 5\k$


%% PROBLEM
Find the equation of the line tangent to the curve
$\r(t) = (t^3+1)\i - (t-1)^2\j + e^{3t-3}\k$ at the point $(2,0,1)$.
%% CORRECT
$x = 2 + 3t$, $y =0$, $z = 1 + 3t$
%% WRONG
$x = 3t$, $y = 0$, $z = 3t$
%% WRONG
$x = 3t^2$, $y =-2(t-1)$, $z = 3e^{3t-3}$
%% WRONG
$x = 2 + 3t^3$, $y = -2(t-1)t$, $z = 1 + 3e^{3t-3}t$
%% WRONG
$x = 2 + 3t$, $y = -2(t-1)$, $z = 1$


%% PROBLEM
A particle's velocity is given by
$\v(t) = e^{-t}\i + e^t\j + 2t\k$, $t \ge 0$.
If the particle is initially at the point $(2,1,1)$, where is it at
time $t = 1$?
%% CORRECT
$(3-e^{-1}, e, 2)$
%% WRONG
$(-e^{-1}, e, 1)$
%% WRONG
$(2-e^{-1}, 1 + e, 2)$
%% WRONG
$(-e^{-t}, e^t, t^2)$
%% WRONG
$(2-e^{-t},1 + e^t, 1 + t^2)$


%% PROBLEM
Calculate the length of the curve $\r(t) = 3t\i + t^2\j +
{4\sqrt{3}\over3}t^{3/2}\k$ from $t = 0$ to $t = 2$.
%% CORRECT
$10$
%% WRONG
$8$
%% WRONG
$6$
%% WRONG
$4$
%% WRONG
$1$


%% PROBLEM
Suppose a particle's position is given by $\r(t) = \sin(t)\i + t \j + t^2\k$.
The tangential component of the particle's acceleration $\a(t) =
a_{\T}(t)\T(t) + a_{\N}(t)\N(t)$ is
$$a_{\T}(t) = {4t - \cos(t)\sin(t)\over \sqrt{\cos^2(t) + 4t^2 + 1}}$$
Use this to find the normal component of the acceleration $a_{\N}(t)$
when $t = \pi$.
%% CORRECT
$a_{\N}(\pi) = {2\over\sqrt{1+2\pi^2}}$
%% WRONG
$a_{\N}(\pi) = 2\k$
%% WRONG
$a_{\N}(\pi) = {4\pi\over\sqrt{2+4\pi^2}}$
%% WRONG
$a_{\N}(\pi) = 4$
%% WRONG
$a_{\N}(\pi) = \sqrt{2 + 4\pi^2}$

%% PROBLEM
Compute the limit $\lim_{(x,y)\to(1,0)} {(x-1)^2y\over (x-1)^2 + y^2}$.
%% CORRECT
$0$
%% WRONG
$\infty$
%% WRONG
{\sl does not exist}
%% WRONG
$1$
%% WRONG
${3\over2}$

%% PROBLEM
Determine which of the functions below has the following level curves:
$$\epsfysize=2truein\epsfbox{exam1.eps}$$
%% CORRECT
$f(x,y) = x^2 - y^2$
%% WRONG
$f(x,y) = x^2 + y^2$
%% WRONG
$f(x,y) = y - x^2$
%% WRONG
$f(x,y) = x - y^2$
%% WRONG
$f(x,y) = x y$

%% END
\end
