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

%% TITLE
\noindent
{\bf Math 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\quad{\sl February 15, 1996}\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 16 multiple choice questions worth 5 points each. You start
with 20 points.

\def\answerspace{0.25in}

%1
%% PROBLEM
Suppose the angle between two unit vectors $\a$ and $\b$ is $30^\circ$.
Compute the dot product of $\a$ and $\b$.

%% CORRECT
${\sqrt{3}\over 2}$
%% WRONG
${1\over 2}$
%% WRONG
${\sqrt{2}\over 2}$
%% WRONG
${\sqrt{3}\over 2} \i + {1\over 2} \j$
%% WRONG
${\sqrt{3}\over 2} \i - {1\over 2} \j$


%2
%% PROBLEM
Find a vector perpendicular to the vectors $\a = 3\i - 2\j + \k$ and
$\b = -\i + 5\j + \k$.

%% CORRECT
$-7\i - 4\j + 13\k$
%% WRONG
$-3\i - 10\j + \k$
%% WRONG
$3\i + 2\j + 17\k$
%% WRONG
$-{4\over 9}\i -{20\over 9}\j -{4\over 9}\k$
%% WRONG
$-{18\over 7}\i +{12\over 7}\j -{6\over 7}\k$


%3
%% PROBLEM
Compute the distance from the point $(2,-1,1)$ to the line $x =1-t$,
$y = 3t$, $z =4+2t$.

%% CORRECT
1.96
%% WRONG
1.62
%% WRONG
1.39
%% WRONG
2.15
%% WRONG
1.05


%4
%% PROBLEM
Determine the distance of the plane $4x - 5y + 2z = 6$ to the origin.

%% CORRECT
0.89
%% WRONG
2.11
%% WRONG
0.54
%% WRONG
1.27
%% WRONG
1.71


%5
%% PROBLEM
Find the volume of the box (parallelepiped) with vertices
$$\vbox{\halign{$#$,\ &$#$,\ &$#$,\ &$#$\cr
(0,0,0) & (3,1,1) & (1,2,4) & (-1,5,5),\cr
(4,3,5) & (2,6,6) & (0,7,9) & (3,8,10).\cr
}}$$

%% CORRECT
32
%% WRONG
30
%% WRONG
7
%% WRONG
19
%% WRONG
24


%6
%% PROBLEM
Which of the following sets of parametric equations describes
the line in which the planes $x - 3y + 5z = 10$ and $x + y - z = 2$
intersect?

%% CORRECT
$x = 4 - t,\quad y = -2 + 3t,\quad z = 2t$
%% WRONG
$x =  -2t,\quad y = 6t,\quad z = 4t$
%% WRONG
$x = 4 + {1\over 2}t,\quad y = 2 + {3\over 2}t,\quad z = t$
%% WRONG
$x = 2 - 2t,\quad y = -1 - 6t,\quad z = 1 + 4t$
%% WRONG
$x = 1 + t,\quad y = 1 - 3t,\quad z = -1 + 5t$


%7
%% PROBLEM
Determine the points in which the curve defined by
$$\r(t) = (t - t^2)\i + t\j + (t+t^2)\k$$
intersects the plane $x-7y+2z=5$.

%% CORRECT
$(-20,5,30)$, $(-2,-1,0)$
%% WRONG
$(20,-5,25)$, $(2,1,5)$
%% WRONG
$(0,0,0)$, $(0,1,3)$
%% WRONG
$(-2,2,-21/2)$, $(-6,-2,-3/2)$
%% WRONG
$(-6,3,16)$, $(-12,-1,5)$


%8
%% PROBLEM
Determine which of the following curves is not smooth at some point in
its domain.

%% CORRECT
$\r(t) = (2t^3-3t^2)\i + (t^3-3t)\j + (t^2-2t)\k$
%% WRONG
$\r(t) = (2t^3+3t^2)\i + (t^3+3t)\j + (t^2+2t)\k$
%% WRONG
$\r(t) = (2t-t^2)\i + (t^3-1)\j + (t-1)^5\k$
%% WRONG
$\r(t) = \cos(t)\i + \sin(t)\j + t^2\k$
%% WRONG
$\r(t) = (t-e^t)\i + (t+e^{-t})\j + (t^3-3t)\k$


%9
%% PROBLEM
Calculate the unit normal vector $\N(1)$ of a curve at $t=1$ given that
its unit tangent vector is
$$\T(t) = {1\over 1 + t}\i + {\sqrt{2t}\over 1+t}\j + {t\over 1 + t}\k$$

%% CORRECT
$-{1\over\sqrt{2}}\i + {1\over\sqrt{2}}\k$
%% WRONG
$-{1\over 4}\i +  {1\over 2\sqrt{2}}\j + {1\over 4}\k $
%% WRONG
$-{1\over(1+t)^2}\i + {1-t\over \sqrt{2}(1+t)^2}\j - {1\over(1+t)^2}\k$
%% WRONG
$-{1\over 4}\i - {1\over 4}\k$
%% WRONG
$-{1\over(1+t)^2}\i + {1\over \sqrt{2t}(1+t)^2}\j + {1\over(1+t)^2}\k$


%10
%% PROBLEM
Suppose a particle initially at rest has acceleration given by $\a(t) =
e^t\i + e^{-t}\j + 2t\k$ at time $t\ge 0$. Compute its {\sl speed} at
time $t = 1$.

%% CORRECT
2.09
%% WRONG
1.65
%% WRONG
0.73
%% WRONG
1.38
%% WRONG
2.92

%11
%% PROBLEM
Find the equation of the plane perpendicular to the curve defined by
$\r(t)  = t^3\i - t^2\j + \sin(\pi t)\k$ at the point $(8,-4,0)$.
%% CORRECT
$3x - y + {\pi\over 4} z = 28$
%% WRONG
$2x - y  = 20$
%% WRONG
$3x + 2y  = 16$
%% WRONG
$3(x-8) - 2(y+4) - \pi z = 0$
%% WRONG
$3t^2 x - 2t y + \pi\cos(\pi t) z = 0$


%12
%% PROBLEM
Compute the limit $\lim_{(x,y)\to(0,0)} {x^2y^3 + x^3y^2\over x^5 + y^5}$.
%% CORRECT
{\sl does not exist}
%% WRONG
{\sl exists, but has many possible values}
%% WRONG
$0$
%% WRONG
$1$
%% WRONG
${2\over5}$


%13
%% PROBLEM
Suppose a particle's position at time $t\ge 0$ is described by
$\r(t) = t\cos(t)\i + t\sin(t)\j + t^2\k$. Calculate the tangential
component of the particle's acceleration.

%% CORRECT
${5t\over\sqrt{1 + 5t^2}}$
%% WRONG
${4t\over\sqrt{2 + 4t^2}}$
%% WRONG
$-\sin(t)\i- \cos(t)\j + 2t\k\over\displaystyle\sqrt{1 + 4t^2}$
%% WRONG
${1\over\sqrt{5}}(-\cos(t)\i - \sin(t)\j + 2\k)$
%% WRONG
${1\over\sqrt{1 + 5t}}$


%14
%% PROBLEM
Which of the following represents the curve traced by
$\r(t) = t\i + \sin(t)\j - t\k$, $0 \le t \le 4$?
%% CORRECT
$\epsfysize=2truein\epsfbox{exam1.1.eps}$
%% WRONG
$\epsfysize=2truein\epsfbox{exam1.2.eps}$
%% WRONG
$\epsfysize=2truein\epsfbox{exam1.3.eps}$
%% WRONG
$\epsfxsize=2.5truein\epsfbox{exam1.4.eps}$
%% WRONG
$\epsfxsize=2.5truein\epsfbox{exam1.5.eps}$


%15
%% PROBLEM
Which of the following integrals gives the length of the curve
$\r(t) = (t^2-2t)\i + 4t\j + (t^2+2t)\k$, $0\le t \le1$?

%% CORRECT
$2\sqrt{2}\int_{0}^1\sqrt{t^2 + 3}\,dt$
%% WRONG
$\sqrt{2}\int_{0}^1t\sqrt{t^2 + 12}\,dt$
%% WRONG
$2\int_{0}^1\sqrt{t + 1}\,dt$
%% WRONG
$\sqrt{2}\int_{0}^1\sqrt{t(t + 2)}\,dt$
%% WRONG
$4\int_{0}^1t + 1\,dt$


%16
%% PROBLEM
Find the equation of the line tangent to the curve $\r(t) = (1 - t^2)\i
+ t^3\j + (1 +t^4)\k$ at the point $(0,1,2)$.

%% CORRECT
$x = -2t,\quad y = 1 + 3t,\quad z = 2 + 4t$
%% WRONG
$x = -2t,\quad y = 3t^2,\quad z = 4t^3$
%% WRONG
$x = 1-2t,\quad y = 3t,\quad z = 1 + 4t$
%% WRONG
$x = -2t,\quad y = 1 + 3t^2,\quad z = 2 + 4t^3$
%% WRONG
$x = t,\quad y = 1,\quad z = 2 + t$

%% END
\bye
