\input epsf
\input calc
\def\problemspace{0.2in}

%% TITLE
\noindent
{\bf Math 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\quad{\sl September 24, 1992}\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
Calculate the distance between the points $(2,1,-1)$ and $(5,3,-7)$.
%% CORRECT
$7$
%% WRONG
$\sqrt{11}$
%% WRONG
$\sqrt{62}$
%% WRONG
$\sqrt{77}$
%% WRONG
$6$



%% PROBLEM
Find the area of the parallelogram with vertices at the points
$(0,0,0)$, $(3,1,2)$, $(1,-1,4)$, and $(4,0,6)$.
%% CORRECT
$12.329$
%% WRONG
$11.543$
%% WRONG
$13.972$
%% WRONG
$10.085$
%% WRONG
$14.151$


%% PROBLEM
Which of the following vectors is perpendicular to  $\v = 3\i - 2\j + 5\k$.
%% CORRECT
$3\i + 2\j - \k$
%% WRONG
$2\i - 3\j$
%% WRONG
$5\i + 3\k$
%% WRONG
$\i + \j - \k$
%% WRONG
$\i - \j - 6\k$


%% PROBLEM
Find the projection of the vector
$v = 5\i - \j + 3\k$ on the vector $w = \i + \j - \k$.
%% CORRECT
${1\over3}\i + {1\over3}\j - {1\over3}\k$
%% WRONG
$1$
%% WRONG
$\i + \j - \k$
%% WRONG
$5\i - \j + 3\k$
%% WRONG
${5\over3}\i - {1\over3}\j + \k$


%% PROBLEM
Let $\a = -3\i + \j + 4\k$ and $\b = 4\i - \k$. Compute $\a \times \b$.
%% CORRECT
$-\i + 13\j - 4\k$
%% WRONG
$-16$
%% WRONG
$-12\i - 4\k$
%% WRONG
$4\i +16\j - \k$
%% WRONG
$3\i +13\j - \k$


%% PROBLEM
Compute the volume of the box (parallelepiped) determined by the vectors
$\a = \i + \j + \k$, $\b = 2\j - \k$, and $\c = 3\i + \j$.
%% CORRECT
$8$
%% WRONG
$3$
%% WRONG
$7$
%% WRONG
$6$
%% WRONG
$12$


%% PROBLEM
Find the point where the line $x = -7 + 2t$, $y = 8 - t$, $z = 3 + 5t$ intersects the plane $x - y + 2z = 4$.
%% CORRECT
$(-5,7,8)$
%% WRONG
$(-3,6,13)$
%% WRONG
$(-9,9,11)$
%% WRONG
$(-11,10,-2)$
%% WRONG
$(1,4,7)$


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


%% PROBLEM
Calculate the distance from the point $(0,10,0)$ to the plane
$x + y - z = 0$.
%% CORRECT
$5.77$
%% WRONG
$10.00$
%% WRONG
$6.84$
%% WRONG
$9.67$
%% WRONG
$11.21$


%% PROBLEM
Determine the equation of the plane through the point $(2,1,0)$ that is parallel to the vectors
$(-1,-1,2)$ and $(-2,1,1)$.
%% CORRECT
$x + y + z = 3$
%% WRONG
$x - y - z = 1$
%% WRONG
$3x + 3y + z = 9$
%% WRONG
$-3x + y - 3z = -5$
%% WRONG
$3x - 3y + z = 3$


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


%% PROBLEM
Which of the following represents the curve traced by
$\r(t) = t\i + 2\sin(t)\j + 3\cos(t)\k$, $0 \le t \le 6\pi$?
%% CORRECT
$\epsfxsize=2truein\epsfbox{exam1.1.eps}$
%% WRONG
$\epsfxsize=2.5truein\epsfbox{exam1.2.eps}$
%% WRONG
$\epsfxsize=2.5truein\epsfbox{exam1.3.eps}$
%% WRONG
$\epsfxsize=2.5truein\epsfbox{exam1.4.eps}$
%% WRONG
$\epsfxsize=2.5truein\epsfbox{exam1.5.eps}$


%% PROBLEM
A particle's velocity is given by
$\v(t) = -t^2\i + t\j + \k$, $t \ge 0$.
If the particle is at the point $(1,2,0)$ at time $t = 1$, where is it at time $t = 2$?
%% CORRECT
$(-{4\over3}, {7\over2}, 1)$
%% WRONG
$(-{8\over3}, 2, 2)$
%% WRONG
$({2\over3}, {5\over2}, 1)$
%% WRONG
$(-{5\over3}, 4, 2)$
%% WRONG
$(-{1\over3}, {1\over2}, 0)$


%% PROBLEM
Calculate the length of the curve $\r(t) = 3t\i + t^2\j + {4\over3}\sqrt{3} t^{3/2}\k$ from $t = 0$ to $t = 3$.
%% CORRECT
$18$
%% WRONG
$16$
%% WRONG
$14$
%% WRONG
$12$
%% WRONG
$9$


%% PROBLEM
Find the unit normal vector $\N(t)$ to the curve
$\r(t) = 2\cos(t)\i - 2\sin(t) \j + \sqrt{5}t\k$.
%% CORRECT
$\N(t) = -\cos(t) \i + \sin(t)\j$
%% WRONG
$\N(t) = -{2\over3}\cos(t) \i + {2\over3}\sin(t)\j + {\sqrt{5}\over3}\k$
%% WRONG
$\N(t) = -{2\over3}\sin(t) \i - {2\over3}\cos(t)\j + {\sqrt{5}\over3}\k$
%% WRONG
$\N(t) = {-2\sin(t) \i - 2\cos(t)\j\over\sqrt{2\sin(t) + 2\cos(t) + \sqrt{5}}}$
%% WRONG
$\N(t) = {-2\cos(t) \i - 2\sin(t)\j + \sqrt{5}\k\over\sqrt{2\cos(t) + 2\sin(t) + \sqrt{5}}}$

%% END
\end
