\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 26, 1991}\hfill
Score:\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 17 multiple
choice questions worth 6 points each.

\def\answerspace{0.25in}

%1
%% PROBLEM
Find a unit vector parallel to the line
$x = -1 + 2t, \quad y = 2 - t, \quad z = 3t$.
%% CORRECT
$0.53\i - 0.27\j + 0.80\k$
%% WRONG
$-0.45\i + 0.89\j$
%% WRONG
$0.58\i- 0.58 \j + 0.58\k$
%% WRONG
$-0.71\i + 0.71\j$
%% WRONG
$\k$



%2
%% PROBLEM
Find a vector perpendicular to the plane
$2x - 3y + 5z = 27$.
%% CORRECT
$2\i - 3\j + 5\k$
%% WRONG
$3\i - 2\j$
%% WRONG
$5\i + 5\j + \k$
%% WRONG
${1\over\sqrt{38}}(\i - \j + \k)$
%% WRONG
${1\over\sqrt{3}}(\i + \j + \k)$




%3
%% PROBLEM
Compute the angle in radians between the vectors $\a = \i - \j + \k$ and $\b = 2\i + \j$.
%% CORRECT
1.31
%% WRONG
1.24
%% WRONG
1.57
%% WRONG
2.03
%% WRONG
1.87





%4
%% PROBLEM
Determine the projection of the vector $\a = 2\i + \j + \k$ onto the vector $\b
= \i + \j + \k$
%% CORRECT
$1.33\i + 1.33\j + 1.33\k$
%% WRONG
$2.68\i + 0.67\j - 0.67\k$
%% WRONG
$0.67\i + 0.67\j + 0.67\k$
%% WRONG
$5.32\i + 1.33\j- 1.33\k$
%% WRONG
$0.58\i + 0.58\j + 0.58\k$



%5
%% PROBLEM
Find the area of the parallelogram with vertices
$(0,0)$, $(3,1)$, $(2,4)$, $(5,5)$.
%% CORRECT
10
%% WRONG
5
%% WRONG
$\sqrt{12}$
%% WRONG
15
%% WRONG
$5\sqrt{5}$



%6
%% PROBLEM
Determine the parametric equations of the line through the points
$(1,-1,1)$ and $(2,0,5)$.
%% CORRECT
$x = 1 + t,\quad y = -1 + t,\quad z = 1 + 4t$
%% WRONG
$x = 1 + 2t,\quad y = -1,\quad z = 1 + 5t$
%% WRONG
$x = 2 + t,\quad y = -t,\quad z = 5 + t$
%% WRONG
$x = 1 + 3t,\quad y = -1 - t,\quad z = 1 + 6t$
%% WRONG
$x = 1 + t,\quad y = 1 - t,\quad z = -4 + 5t$



%7
%% PROBLEM
Compute the distance from the point $(2,1,1)$ to the line $x = t,\quad y = 2t,\quad z = 3t$.
%% CORRECT
1.58
%% WRONG
1.32
%% WRONG
1.79
%% WRONG
2.05
%% WRONG
2.25



%8
%% PROBLEM
Find the equation of the plane through the origin that is parallel to the  vectors $\a = \i +
\j + \k$ and $\b = \i$.
%% CORRECT
$y - z = 0$
%% WRONG
$x + y + z = 0$
%% WRONG
$x + y - z = 0$
%% WRONG
$x - y = 0$
%% WRONG
$x -z = 0$



%9
%% PROBLEM
Determine the distance of the plane
$3x - y + 2z = 6$
to the origin.
%% CORRECT
1.6
%% WRONG
2.0
%% WRONG
0.8
%% WRONG
1.2
%% WRONG
1.0



%10
%% PROBLEM
Let $\r(t) = e^t\i + e^{-t}\j + t\k$.
Compute $\r'(0)$.
%% CORRECT
$\i - \j + \k$
%% WRONG
$\i - \j$
%% WRONG
$e^t\i - e^{-t} \j + t\k$
%% WRONG
$e^t\i + e^{-t} \j + \k$
%% WRONG
$\i + \j + \k$




%11
%% PROBLEM
Which of the following curves is not smooth at some point in its domain?
%% CORRECT
$\r(t) = t^2\i + t^3\j + t^5\k$
%% WRONG
$\r(t) = t\i + t^2\j + t^3\k$
%% WRONG
$\r(t) = (t+t^2)\i + t^3\j + (t-1)^5\k$
%% WRONG
$\r(t) = \cos(t)\i + \sin(t)\j + t\k$
%% WRONG
$\r(t) = e^t\i + e^{-t}\j + t\k$




%12
%% PROBLEM
Find a parameterization of
the graph of the function $f(x) = x^3 - x$.
%% CORRECT
$\r(t) = t\i + (t^3 - t)\j$
%% WRONG
$\r(t) = (t^2 - 1)\i + t\j$
%% WRONG
$\r(t) = (t - 1)\i + t(t+1)\j$
%% WRONG
$\r(t) = t\i + t^3\j - t\k$
%% WRONG
$\r(t) = t\i + (t^2-1)\j$





%13
%% PROBLEM
Suppose a particle initially at rest has the following acceleration
$\a(t) = t\i + t^2\j + t^3\k$, $t\ge 0$.
Calculate the particle's speed at $t = 1$.
%% CORRECT
0.65
%% WRONG
0.45
%% WRONG
0.85
%% WRONG
1.05
%% WRONG
1.25



%14
%% PROBLEM
Suppose $\F(t) = e^t\i + t\j - {1\over t}\k$. Compute $\int_1^2 \F(t)\,dt$.
%% CORRECT
$4.67\i + 1.50\j - 0.69\k$
%% WRONG
$7.39\i + 2.00\j - 0.69\k$
%% WRONG
$4.67\i + 0.75\j$
%% WRONG
$8.70\j$
%% WRONG
$2.72\i + 1.00\j - 0.25\k$



%15
%% PROBLEM
Which of the following integrals gives the length of the curve
$\r(t) = t\cos(t)\i + t\sin(t)\j + t^2\k$, $0\le t \le \pi$?
%% CORRECT
$\int_0^\pi\sqrt{1 + 5t^2}\,dt$
%% WRONG
$\int_0^\pi\sqrt{t(\cos(t) + \sin(t) + t)}\,dt$
%% WRONG
$\int_0^\pi\sqrt{\sin(t) - \cos(t) + 2t}\,dt$
%% WRONG
$\int_0^\pi\sqrt{1 + t^2}\,dt$
%% WRONG
$\int_0^\pi2t\,dt$






%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
$x = t,\quad y = 1,\quad z = 2 + t$




%17
%% PROBLEM
Suppose a particle's motion 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
${1\over\sqrt{1 + 5t}}$





%% END
\bye

