\input calc
\def\ds{\displaystyle}

%% TITLE
{\bf Math 225: Calculus III}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam I}\quad{\sl September 27, 1990}\hfill Score:\hbox to 2.5truein{\hrulefill}

\bigskip\bigskip
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 for a total of 102 points.



%1
%% PROBLEM
The distance between the points $(1, -2, 5)$ and $(-1, 6, 3)$ is best
approximated by
%% CORRECT
{$8.49$}
%% WRONG
{$4.45$}
%% WRONG
{$6.23$}
%% WRONG
{$2.57$}
%% WRONG
{$10.14$}

%2
%% PROBLEM
Which two vectors are perpendicular?
%% CORRECT
{$3\i - \j -2\k$ and $-\i +\j -2\k$}
%% WRONG
{$-\i + 2\j + \k$ and $\i - 2\j + \k$}
%% WRONG
{$\i - \j$ and $\j - \k$}
%% WRONG
{$\i$ and $\i + \j + \k$}
%% WRONG
{$\i +\j + 2\k$ and $3\i + 3\j - 6\k$}

%3
%% PROBLEM
Find the angle between the vectors $\a = \i + \j + \k$ and
$\b = (\sqrt{2}+3)\i + \sqrt{2}\j + (\sqrt{2}-3)\k$.
%% CORRECT
{$\pi/3$}
%% WRONG
{$\pi/6$}
%% WRONG
{0}
%% WRONG
{$\pi/2$}
%% WRONG
{$3\pi/4$}

%4
%% PROBLEM
Find the projection of the vector $\a = \i + 2\j$ on the vector $\b = -\i
+ \j +\k$.
%% CORRECT
{${1\over3}(-\i + \j + \k)$}
%% WRONG
{${1\over\sqrt{15}}(-\i + \j + \k)$}
%% WRONG
{${1\over\sqrt{5}}(\i + 2\j)$}
%% WRONG
{$-\i + \j + \k$}
%% WRONG
{$\bf 0$}

%5
%% PROBLEM
Compute the triple product $\a\times(\b\times\c)$ of the vectors
$\a = \i + \j$, $\b = \j + \k$, and $\c = \i  + \k$.
%% CORRECT
{$\j-\i$}
%% WRONG
{$\0$}
%% WRONG
{$\k-\i$}
%% WRONG
{$\k-\j$}
%% WRONG
{$\i-\j + \k$}

%6
%% PROBLEM
Which of the following vectors is normal to the plane containing the points
$P_1 = (5,0,1)$, $P_2 = (1,1,0)$, and $P_3 (-1,0,0)$?
%% CORRECT
{$\i - 2\j - 6\k$}
%% WRONG
{$\i - 4\j + 3\k$}
%% WRONG
{$-\i - \j + 5\k$}
%% WRONG
{$\i - \j + 5\k$}
%% WRONG
{$\i + 2\j - 2\k$}

%7
%% PROBLEM
Find the distance from the point $(0,1,0)$ to the line
$x=1$, ${y\over 4} = {(z - 2)\over 3}$.
%% CORRECT
{$\sqrt{146}\over\ds 5$}
%% WRONG
{$\sqrt{159}\over\ds 5$}
%% WRONG
{$\sqrt{212}\over\ds 4$}
%% WRONG
{$\sqrt{116}\over\ds 3$}
%% WRONG
{$\sqrt{189}\over\ds 4$}

%8
%% PROBLEM
Find the distance from the point $(1,2,-3)$ to the plane
$7x + 3y + z = 5$.
%% CORRECT
{$5\over \sqrt{\ds59}$}
%% WRONG
{$10\over \sqrt{\ds59}$}
%% WRONG
{$3\over \sqrt{\ds13}$}
%% WRONG
{$25\over \sqrt{\ds767}$}
%% WRONG
{$5\over \sqrt{\ds13}$}

%9
%% PROBLEM
Find the point where the plane
$2x -3y - z = 6$ and the line
$\r(t) = (1+t)\i + t\j + (1-t)\k$ intersect.
%% CORRECT
{{\sl no such point}}
%% WRONG
{$(1,-1,-1)$}
%% WRONG
{$(1,0,1)$}
%% WRONG
{$(2,1,0)$}
%% WRONG
{$(0,0,-6)$}

%10
%% PROBLEM
Determine the symmetric equations of the line tangent to the curve
$\r(t) = \cos(t)\i + \sin(t)\j + \sin(2t)\k$ at the point $(1,0,0)$.
%% CORRECT
{$x=1,\quad y = {z\over2}$}
%% WRONG
{$x-1 = y = {z\over2}$}
%% WRONG
{$x$-axis}
%% WRONG
{$x=1,\quad y = 0$}
%% WRONG
{$y=0,\quad x-1 = {z\over 2}$}

%11
%% PROBLEM
Which of the following statements applies to
the curve defined by $\r(t) = t^2\i + t^3\j + t^5\k$.
%% CORRECT
{{\sl piecewise smooth, but not smooth}}
%% WRONG
{{\sl smooth}}
%% WRONG
{{\sl continuous, but not piecewise smooth}}
%% WRONG
{{\sl not continuous}}
%% WRONG
{{\sl none of the above apply}}

%12
%% PROBLEM
The position of a particle at time $t\ge 0$ is given by
$\r(t) = (t^3-t)\i + t^2\j + (t^5-t^3)\k$.
Compute the {\sl speed} of the particle at time $t = 1$.
%% CORRECT
{$2\sqrt{3}$}
%% WRONG
{$2\i + 2\j + 2\k$}
%% WRONG
{$\sqrt{25t^8 - 30t^6 + 18t^4 - 2t^2}$}
%% WRONG
{$3\i + 2\j$}
%% WRONG
{$\sqrt{11}$}

%13
%% PROBLEM
The tangential component of
acceleration of an object with position
$\r(t) = e^t\i + e^{-t}\j + \sqrt{2}\,t\,\k$
is $a_T = e^t - e^{-t}$. Determine the normal component of
acceleration,~$a_N$.
%% CORRECT
{$\sqrt{2}$}
%% WRONG
{$\sqrt{e^{2t} + e^{-2t}} $}
%% WRONG
{$e^t + e^{-t}$}
%% WRONG
{$e^t\i + e^{-t}\j$}
%% WRONG
{$e^t\i - e^{-t}\j + \sqrt{2}\k$}

%14
%% PROBLEM
Suppose $\F(0) = \i + \j + 2\k$, $\F'(0) = \j-\k$, $\G(0) = -5\i - 2\j + 3\k$,
and $\G'(0) = \i+ 4\j$. Compute $(\F\bullet\G)'(0)$.
%% CORRECT
{$0$}
%% WRONG
{$-5$}
%% WRONG
{$5$}
%% WRONG
{$-4$}
%% WRONG
{$1$}

%15
%% PROBLEM
Let $\F(t) = t\sin(t)\i + t\cos(t)\j + te^t\k$.
Find $\F'(0)$.
%% CORRECT
{$\j + \k$}
%% WRONG
{$(\sin(t) + t\cos(t))\i + (\cos(t)-t\sin(t))\j + e^t\k$}
%% WRONG
{$\i + \k$}
%% WRONG
{$\cos(t) + t$}
%% WRONG
{$2$}

%16
%% PROBLEM
Determine a vector valued function $\F(t)$ such that
$\F'(t) = \i + 2e^{-t}\j + {1\over 1+t}\k$ and
$\F(0) = \i$.
%% CORRECT
{$\F(t) = (t + 1)\i + 2(1-e^{-t})\j + \ln(1+t)\k$}
%% WRONG
{$\F(t) = (1 - t)\i + (1-e^{-t})\j + \ln(1+t)\k$}
%% WRONG
{$\F(t) = t \i +(2-e^{-t})\j -(1+\ln(1+t))\k$}
%% WRONG
{$\F(t) = t\i - 2e^{-t}\j + \ln(1+t)\k$}
%% WRONG
{$\F(t) = (t^2 + 1)\i + (1-e^{-2t})\j - \ln(1+t)\k$}

%17
%% PROBLEM
Which of the following is the length of the curve parameterized
by  $\r(t) = (1+t)\i + (2 - t^2)\j + (3 + t^3)\k$, $0\le t \le 1$.

%% CORRECT
{$\int_0^1\sqrt{1 + 4t^2 + 9t^4}\, dt$}
%% WRONG
{$52\over\ds 15$}
%% WRONG
{${7\over 3}$}
%% WRONG
{$\sqrt{14}$}
%% WRONG
{$\int_0^1\sqrt{14 + 2t - 7t^2 + 4t^4 + 18t^3 + 9t^6}\, dt$}

%% END
\bye
