\input epsf
\input calc
\def\problemspace{0.5in}
\def\ds{\displaystyle}

%% TITLE
\noindent
{\bf Math 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\quad{\sl February 3, 1994}\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 6 points each. You start with 10 points.

\def\answerspace{0.25in}


%% PROBLEM
Particle $a$ has position vector $\a(t) = t^2\i + t\j - \k$ and particle
$b$ has position vector $\b(t) = t\i - t^3\j + t\k$. Determine how fast
$b$ appears to be moving from $a$'s point of view at time $t = 1$?
%% CORRECT
$3\sqrt{2}$
%% WRONG
$2\sqrt{3}$
%% WRONG
$5$
%% WRONG
$\sqrt{4t^2 + 1}$
%% WRONG
$\sqrt{5}$



%% PROBLEM
Find the area of the triangle with vertices at the points
$(0,0,0)$, $(1,1,2)$, and $(1,-1,4)$.
%% CORRECT
$\sqrt{11}$
%% WRONG
${11\over2}\sqrt{2}$
%% WRONG
$8$
%% WRONG
$4$
%% WRONG
$\sqrt{2}$


%% PROBLEM
Find the point where the line $x = 2 - t$, $y = 1 - 2t$, $z =-2 + 3t$,
intersects the plane $x - y + z = 3$.
%% CORRECT
$(1,-1,1)$
%% WRONG
$({1\over2}, {3\over2}, {3\over2})$
%% WRONG
$({7\over4}, {1\over2}, -{5\over4})$
%% WRONG
$({5\over4}, -{3\over2}, {1\over4})$
%% WRONG
$(2, 1, -2)$


%% PROBLEM
Find the projection of the vector
$\v = 2\i - 4\j + \k$ on the vector $\w = \i + \j -\k$.
%% CORRECT
$-\i - \j + \k$
%% WRONG
$2\i - 4\j - \k$
%% WRONG
${1\over7}\i + {1\over7}\j - {1\over7}\k$
%% WRONG
$-2\i + 4\j - \k$
%% WRONG
${2\over7}\i - {4\over7}\j + {1\over7}\k$


%% PROBLEM
Find the equation of the plane perpendicular to the line
$x = 7 - 3t$, $y = 5 + 4t$, $z = -3 - t$ through the point $(1,0,1)$.
%% CORRECT
$3x - 4y + z = 4$
%% WRONG
$3x - 4y + z = 0$
%% WRONG
$-3x + 4y - z = 6$
%% WRONG
$-3(x-7) + 4(y-5) - (z+3) = 0$
%% WRONG
$3x + 4y - z = 9$


%% PROBLEM
Compute the volume of the box (parallelepiped) determined by the vectors
$\a = 2\i - \j + 3\k$, $\b = 5\i + \j$, and $\c = 2\j - \k$.
%% CORRECT
$23$
%% WRONG
$15$
%% WRONG
$30$
%% WRONG
$60$
%% WRONG
$26$


%% PROBLEM
Find the point on the line $x = -1 + t$, $y = 1 - t$, $z = t$ closest
to the origin.
%% CORRECT
$(-{1\over3},{1\over3},{2\over3})$
%% WRONG
$(0,{1\over2}, {1\over2})$
%% WRONG
$(0,0,1)$
%% WRONG
$(-1,1,0)$
%% WRONG
$({1\over3},{2\over3},{2\over3})$


%% PROBLEM
Calculate the distance from the point $(2,-1,5)$ to the plane
$x + 2y - z = 1$.

%% CORRECT
$\sqrt{6}$
%% WRONG
$1\over\sqrt{\ds 6}$
%% WRONG
$\sqrt{27}$
%% WRONG
$6$
%% WRONG
$1\over\sqrt{\ds27}$

%% PROBLEM
Find a vector perpendicular to the vectors $\a = -3\i+5\j+9\k$ and
$\b=7\i+\k$

%% CORRECT
$5\i+66\j-35\k$
%% WRONG
$4\i+52\j-28\k$
%% WRONG
$-3\i-39\j+21\k$
%% WRONG
$-2\i-26\j+14\k$
%% WRONG
$1\i+13\j-7\k$


%% PROBLEM
Determine the equation of the line tangent to the curve $\r(t) = (1 + t)\i+
(2 + t^2)\j + (3 - t^3)\k$ at the point $(0,3,4)$.
%% CORRECT
$x= t$, $y=3-2t$, $z=4-3t$
%% WRONG
$x=1$, $y=2t$, $z=-3t^2$
%% WRONG
$x=1$, $y=3+2t$, $z=4-3t^2$
%% WRONG
$x= 1+t$, $y=2-2t$, $z=3-3t$
%% WRONG
$x=1+t$, $y=2+2t$, $z=4-3t$

%% PROBLEM
The total force acting on a particle of mass 2 at time $t$ is  given by
$\F(t) = 8e^{2t}\i + 8e^{-2t}\j$. If the particle starts at the origin with
initial velocity $\v_0 = 2\i - 2\j + 3\k$, find the position of the particle
at time $t=1$.

%% CORRECT
$(e^2-1)\i + (e^{-2}-1)\j + 3\k$
%% WRONG
$(e^2+2)\i + (e^{-2}-2)\j + 3\k$
%% WRONG
$(e^2+1)\i + (e^{-2}-3)\j + 3\k$
%% WRONG
$e^2\i + e^{-2}\j + 3\k$
%% WRONG
$(e^2+1)\i - (e^{-2}+1)\j + 3\k$


%% PROBLEM
Determine which of the following curves is not smooth at some point
in its domain.

%% CORRECT
$\r(t) = t^3\i + \cos(t)\j$
%% WRONG
$\r(t) = t^3\i + \sin(t)\j$
%% WRONG
$\r(t) = t^3\i + t^2\j + t\k$
%% WRONG
$\r(t) = (t^3 - 3t^2)\i + (t^2 - 2t)\j + t^2\k$
%% WRONG
$\r(t) = (t-1)^3\i + t^2\j$

%% PROBLEM
Compute the approximate angle in radians bewteen the vectors $\a
= 3\i - j + 2\k$ and $\b = 2\i + 2\j + \k$.

%% CORRECT
$1.01$
%% WRONG
$1.48$
%% WRONG
$1.72$
%% WRONG
$0.79$
%% WRONG
$1.34$

%% PROBLEM
Determine which of the following integrals gives the length of the curve
$$\r(t) = t\cos(\pi t)\i + t\sin(\pi t)\j + \k,\qquad 0 \le t \le 1$$

%% CORRECT
$\int_0^1 \sqrt{1 + \pi^2 t^2}\,dt$
%% WRONG
$\int_0^{1}\sqrt{1+ \pi t}\, dt$
%% WRONG
$\int_0^1 \sqrt{(1+\pi t)\cos(\pi t) + (1-\pi t)\sin(\pi t)}\, dt$
%% WRONG
$\int_0^{1} \sqrt{(1+\pi t)^2 + (1-\pi t)^2}\, dt$
%% WRONG
$\int_0^1 \sqrt{1 + (1+\pi t)^2\cos^2(\pi t) + (1-\pi t)^2 \sin^2(\pi t)}\, dt$


%% PROBLEM
Determine the unit normal vector $\N(t)$ of a curve given that its
unit tangent is $$\T(t) = {t^2\over t^2+1}\i + {\sqrt{2}\,t\over t^2+1}\j
+ {1\over t^2+1}\k$$

%% CORRECT
$\N(t) = {\sqrt{2}\,t\over t^2+1}\i +{1-t^2\over t^2+1}\j -
{\sqrt{2}\,t\over t^2+1}$
%% WRONG
$\N(t) = {2t\over (t^2 + 1)^2}\i + {\sqrt{2}\,(1-t^2)\over (t^2 + 1)^2}\j - 
{2t\over (t^2 + 1)^2}\k$
%% WRONG
$\N(t) = {1\over2}t^2\i+\sqrt{2}\,t\j+\ln(t)\k$
%% WRONG
$\N(t) = {-2t^3\over (t^2+1)^2}\i + {2\sqrt{2}\, t^2\over (t^2+1)^2}\j -
{2t \over (t^2+1)^2}\k$
%% WRONG
$\N(t) = {1\over t^2+1}\i + {\sqrt{2}\,t\over t^2+1}\j
- {t^2\over t^2+1}\k$

%% END
\end
