\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 16, 1995}\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
Let $\a = 2\i - \j + 3\k$ and $\b = -\i + 5\j$. Compute
$\a\times\b$.
%% CORRECT
$-15\i-3\j+9\k$
%% WRONG
$-2\i-5\j$
%% WRONG
$10\i+3\j-\k$
%% WRONG
$-5\i+13\j-6\k$
%% WRONG
$\i+4\j$



%% PROBLEM
Three vertices of a parallelogram are $(1,2)$, $(3,3)$, and $(2,4)$.
Determine which of the following points could be the fourth vertex.
%% CORRECT
$(4,5)$
%% WRONG
$(3,0)$
%% WRONG
$(0,4)$
%% WRONG
$(0,0)$
%% WRONG
$(3,1)$


%% PROBLEM
Compute the angle between the planes $x + y + z = 3$ and $4x - 3y = 1$.
%% CORRECT
$1.455$ radians
%% WRONG
$1.571$ radians
%% WRONG
$0.785$ radians
%% WRONG
$1.047$ radians
%% WRONG
$1.134$ radians


%% PROBLEM
Find the equation of the line that passes through the points
$(3,-1,0)$ and $(4,5,1)$.
%% CORRECT
$x = 3 + t$, $y = -1 + 6t$, $z = t$
%% WRONG
$x = 1 + 3t$, $y = 6 - t$, $z = 0$
%% WRONG
$x = 3 + 4t$, $y = -1 + 5t$, $z = t$
%% WRONG
$x = 4 + 3t$, $y = 5 - t$, $z = 0$
%% WRONG
$x = 3t$, $y = 6t$, $z = t$


%% PROBLEM
Determine which of the vector functions below defines the following curve:
$$\epsfysize=2truein\epsfbox{exam1.eps}$$
%% CORRECT
$\r(t) = \sin(t)\i + t\j + \cos(t)\k$
%% WRONG
$\r(t) = \cos(t)\i - \sin(t)\j + t\k$
%% WRONG
$\r(t) = \cos(t)\i + \sin(t)\j + t\k$
%% WRONG
$\r(t) = t\i + \cos(t)\j + \sin(t)\k$
%% WRONG
$\r(t) = t\i - \sin(t)\j + \cos(t)\k$


%% PROBLEM
Find a vector function $\r(t)$ such that $\r''(t) = e^t\i + e^{-t}\j$,
$\r'(0) = \k$, and $\r(0) = \i - \j$.
%% CORRECT
$\r(t) = (e^t - t)\i + (e^{-t} + t - 2)\j + t\k$
%% WRONG
$\r(t) = (e^t + 1)\i + (e^{-t} - 1)\j + t\k$
%% WRONG
$\r(t) = e^t\i - e^{-t}\j + t\k$
%% WRONG
$\r(t) = (t - e^t)\i + (t - e^{-t})\j + (t-1)\k$
%% WRONG
$\r(t) = (e^t + t)\i + (e^{-t} - t)\j + {1\over2}t^2\k$


%% PROBLEM
The position of a particle at time $t$ is given by
$\r(t) = (t^3 - t)\i + (t^2 - t)\j + 2t\k$.
Find the direction in which the particle is moving at time $t = 1$.
%% CORRECT
$2\i+\j+2\k$
%% WRONG
$3\i+2\j+2\k$
%% WRONG
$(3t^2-1)\i+(2t-1)\j+2\k$
%% WRONG
$6\i+2\j$
%% WRONG
$-{1\over4}\i-{1\over6}\j+\k$


%% PROBLEM
Compute the unit normal vector to the curve defined by
$\r(t) = (\cos(t)+t)\i + (t - \cos(t))\j + \sqrt{2}\sin(t)\k$
%% CORRECT
$\N(t) = -{1\over\sqrt{2}}\cos(t)\i + {1\over\sqrt{2}}\cos(t)\j - \sin(t)\k$
%% WRONG
$\N(t) = -\cos(t)\i + \cos(t)\j - \sqrt{2}\sin(t)\k$
%% WRONG
$\N(t) = {1\over\sqrt{2}}\sin(t)\i - {1\over\sqrt{2}}\sin(t)\j + \cos(t)\k$
%% WRONG
$\N(t) = -\cos(t)\i - \cos(t)\j + \sin(t)\k$
%% WRONG
$\N(t) = {1\over\sqrt{2}}\cos(t)\i + \cos(t)\j - {1\over\sqrt{2}}\sin(t)\k$

%% PROBLEM
Find the tangential component of acceleration, $a_T(t)$, of a particle
whose motion is described by $$\r(t) = (e^t + e^{-t})\i + (e^t - e^{-t})\j$$
%% CORRECT
${\sqrt{2}(e^{2t} - e^{-2t})\over\sqrt{e^{2t} + e^{-2t}}}$
%% WRONG
$\sqrt{2(e^{2t} + e^{-2t})}$
%% WRONG
$(e^t + e^{-t})\i + (e^t - e^{-t})\j$
%% WRONG
$2(e^{2t} + e^{-2t})$
%% WRONG
${(e^t + e^{-t})\over\sqrt{e^{2t} + e^{-2t}}}\i + 
{(e^t - e^{-t})\over\sqrt{e^{2t} + e^{-2t}}}\j$


%% PROBLEM
Determine which of the following integrals gives the length of the curve defined by
$\r(t) = t^2\i + (t^3 - t)\j + t^5\k$, $0 \le t \le 2$.
%% CORRECT
$\int_0^2\sqrt{25t^8 + 9t^4 - 2t^2 + 1}\,dt$
%% WRONG
$\int_0^2 2t\i + (3t^2-1)\j + 5t^4\k\,dt$
%% WRONG
$\int_0^2\sqrt{5t^4 + 3t^2+2t-1}\,dt$
%% WRONG
$\int_0^2\sqrt{t^{10} + t^6 +t^4 - 2t^3 + t^2}\,dt$
%% WRONG
$\int_0^2 {1\over3}t^3\i + ({1\over4}t^4-{1\over2}t^2)\j + {1\over6}t^6\k\,dt$

%% PROBLEM
Compute the distance from the point $(1,3,7)$ to the line
$x = 2 - t$,  $y = 6 + 2t$, $z = 3t$.
%% CORRECT
$6.38$
%% WRONG
$5.27$
%% WRONG
$4.16$
%% WRONG
$7.49$
%% WRONG
$3.05$


%% PROBLEM
Calculate the distance from the point $(-1,-1,1)$ to the plane $x + y + z = 5$.
%% CORRECT
$2\sqrt{3}$
%% WRONG
${5\over\sqrt{3}}$
%% WRONG
$5\sqrt{3}$
%% WRONG
${\sqrt{3}\over2}$
%% WRONG
$\sqrt{6}$

%% PROBLEM
Determine the equations of the line tangent to the curve
$\r(t) = 3t^2\i - t\j + (t^3 -1 )\k$
at the point $(12,-2,7)$
%% CORRECT
$x = 12 + 12t$, $y = -2 - t$, $z = 7 + 12t$
%% WRONG
$x = 12 + 6t$, $y = -3$, $z = 7 + 3t^2$
%% WRONG
$x = 12 + 6t^2$, $y = -3t$, $z = 7 + 3t^3$
%% WRONG
$x = 12 + 6t$, $y = -2 - t$, $z = 7 + 3t$
%% WRONG
$x = 12 + 12t$, $y = -2$, $z = 7 + 6t$

%% PROBLEM
Compute the limit
$\lim_{(x,y)\to(1,1)} {(xy - x + y - 1)^2\over xy^2 - 2xy + x}$
%% CORRECT
$4$
%% WRONG
{\sl exists, but has many possible values}
%% WRONG
$0$
%% WRONG
{\sl does not exist}
%% WRONG
${1\over2}$


%% PROBLEM
Let $f(x,y) = e^{2x}(y^3 - xy) + xy^2$. Compute $f_{xy}(2,5)$.
%% CORRECT
$145e^4+10$
%% WRONG
$75e^4+10$
%% WRONG
$150e^4$
%% WRONG
$2e^4+10$
%% WRONG
$115e^4+25$

%% END
\end
