\input calc
\input epsf
\mathsurround=1pt
\def\problemspace{0.1in}
\def\C{{\cal C}}

%% TITLE
\noindent
{\bf Math 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\quad{\sl May 2, 1994}\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 25 multiple
choice questions worth 5 points each. You start with 25 points.

\def\answerspace{0.25in}


%% PROBLEM
Find the area of the parallelogram with vertices $(0,0,0)$, $(1,2,-1)$,
$(0,1,1)$, and $(1,3,0)$.

%% CORRECT
$3.32$
%% WRONG
$2.87$
%% WRONG
$3.94$
%% WRONG
$2.60$
%% WRONG
$2.35$


%% PROBLEM
Find the parametric equations of the line that passes through the
point $(1,0,-1)$ and is perpendicular to the plane $2x - y + 3z
= 7$.

%% CORRECT
\vtop{\halign{\hfil$#$ &  $#$\hfil\cr
x&=1+2t\cr y&=-t\cr z&=-1+3t\cr}}
%% WRONG
$2(x-1) - y + 3(z+1) = 7$
%% WRONG
\vtop{\halign{\hfil$#$ &  $#$\hfil\cr
x&=2+t\cr y&=-1\cr z&=3-t\cr}}
%% WRONG
\vtop{\halign{\hfil$#$ &  $#$\hfil\cr
x&=2t\cr y&=-t\cr z&=3t\cr}}
%% WRONG
\vtop{\halign{\hfil$#$ &  $#$\hfil\cr
x&=t\cr y&=0\cr z&=-t\cr}}


%% PROBLEM
Find the equation of the plane through the point $(3,1,5)$
perpendicular to the $y$-axis.

%% CORRECT
$y=1$
%% WRONG
$x+z=8$
%% WRONG
$3x+y+5z=9$
%% WRONG
$x+y+z=9$
%% WRONG
$3(x-3)+(y-1)+5(z-5)=0$


%% PROBLEM
A particle's position is given by $\r(t) = \cos(\pi t)\i + \sin(\pi t)\j +
t^2\k$. Find its speed at $t = 2$.

%% CORRECT
$\sqrt{16+\pi^2}$
%% WRONG
$-\pi\i+\pi\j+4\k$
%% WRONG
$\sqrt{17}$
%% WRONG
$2\pi$
%% WRONG
$4\j$


%% PROBLEM
Suppose a particle's acceleration is $\a(t) = e^t\i + e^{-t}\j + \k$.
Find the particle's position at time $t=1$ if it is initially at rest at
the origin.

%% CORRECT
$(e-2)\i+e^{-1}\j + {1\over2}\k$
%% WRONG
$e\i+e^{-1}\j + {1\over2}\k$
%% WRONG
$(e-2)\i+(e^{-1}+2)\j + \k$
%% WRONG
$e\i+(e^{-1}+2)\j + {1\over2}\k$
%% WRONG
$(e-1)\i+(e^{-1}+1)\j + \k$


%% PROBLEM
Let $\r(t) = 2t\i + 3t^2\j - t^3\k$. Find the unit tangent vector,
$\T(t)$, at $t = 1$.
%% CORRECT
${2\over7}\i + {6\over7}\j - {3\over7}\k$
%% WRONG
$2\i + 2\j - 3\k$
%% WRONG
$2\i + 6t\j - 3t^2\k$
%% WRONG
${2\over\sqrt{4 + 4t^2 + 9t^4}}\i + {2t\over\sqrt{4 + 4t^2 + 9t^4}}\j
- {3t^2\over\sqrt{4 + 4t^2 + 9t^4}}\k$
%% WRONG
${2\over\sqrt{17}}\i + {2\over\sqrt{17}}\j - {3\over\sqrt{17}}\k$


%% PROBLEM
Compute $\lim_{(x,y)\to(0,0)} {x^3 + y^3\over x^2+y^2}$.
%% CORRECT
$0$
%% WRONG
$1$
%% WRONG
$2$
%% WRONG
${1\over2}$
%% WRONG
{\sl does not exist}


%% PROBLEM
Let $f(x,y) = xe^{xy}$. Compute $f_{xy}(2,1)$.
%% CORRECT
$8e^2$
%% WRONG
$3e^2$
%% WRONG
$2e^2$
%% WRONG
$4e^2$
%% WRONG
$6e^2$


%% PROBLEM
The position and velocity of a smoothly moving particle at time
$t=1$ are $\r(1) = 3\i+4\j$ and $\v(1) = 5\i-5\j$, respectively.
Determine how fast the distance of the particle to the origin is
changing at time $t=1$.

%% CORRECT
$-1$
%% WRONG
$-5$
%% WRONG
$5\sqrt{2}$
%% WRONG
$-\sqrt{2}/2$
%% WRONG
$35$


%% PROBLEM
Find the direction in which the function $f(x,y) = \sin(x^2 + xy)$ is
increasing most rapidly at $(0,1)$.

%% CORRECT
$\i$
%% WRONG
$\j$
%% WRONG
$\i+\j$
%% WRONG
$\i-\j$
%% WRONG
$2\i+\j$


%% PROBLEM
Find the equation of the plane tangent to the graph of $f(x,y) = x^2y - y^3$
at the point $(2,1)$.

%% CORRECT
$4x+y-z = 6$
%% WRONG
$4x+y-z=0$
%% WRONG
$2xy+x^2-3y^2 = 5$
%% WRONG
$4x+y=9$
%% WRONG
$4x+y=0$


%% PROBLEM
Find all of the critical points of the function $f(x,y) = x^2y - 6xy + 8y -
x^2 + 6x - 8$.

%% CORRECT
$(2,1)$, $(4,1)$
%% WRONG
$(2,1)$, $(3,1)$, $(4,1)$
%% WRONG
$(2,1)$, $(2,-1)$, $(3,1)$, $(3,-1)$, $(4,1)$, $(4,-1)$
%% WRONG
$(2,0)$, $(2,1)$, $(4,0)$, $(4,1)$, $(3,0)$, $(3,1)$
%% WRONG
$(3,1)$


%% PROBLEM
Let $f(x,y) = x^3y - 3x^2y$. determine which of the following statements
is true.

%% CORRECT
$f$ has a saddle point at $(3,0)$.
%% WRONG
$f$ has a local minimum at $(3,0)$.
%% WRONG
$f$ has a local maximum at $(3,0)$.
%% WRONG
$f$ is not continuous at $(3,0)$.
%% WRONG
$(3,0)$ is not a critical point of $f$.


%% PROBLEM
Determine which of the following sysytems of equations must be solved to find
the extrema of the function $f(x,y) = x^2y - y^3$ subject to the constraint
$x^4 + y^4 = 1$ using the method of Lagrange multipliers.

%% CORRECT
\vtop{\halign{\hfil$#$ &  $#$\hfil\cr
xy&=2\lambda x^3\cr x^2-3y^2&=4\lambda y^3\cr x^4+y^4&=1\cr}}
%% WRONG
\vtop{\halign{\hfil$#$ &  $#$\hfil\cr
2xy&=\lambda x^4\cr x^2-3y^2&=\lambda y^4\cr x^4+y^4&=1\cr}}
%% WRONG
\vtop{\halign{\hfil$#$ &  $#$\hfil\cr
xy&=0\cr x^2-3y^2&=0\cr x^3 + y^3&=0\cr}}
%% WRONG
\vtop{\halign{\hfil$#$ &  $#$\hfil\cr
xy&=0\cr x^2-3y^2&=0\cr x^4+y^4&=1\cr}}
%% WRONG
\vtop{\halign{\hfil$#$ &  $#$\hfil\cr
xy&=4\lambda x^3\cr x^2-3y^2&=4\lambda y^3\cr x^4+y^4&=0\cr}}


%% PROBLEM
Find the area enclosed by one leaf of the rose $r = 6\sin(2\theta)$.
$$\epsfxsize=1truein\epsfbox{final.1.eps}$$

%% CORRECT
$9\pi/2$
%% WRONG
$18\pi$
%% WRONG
$9\pi$
%% WRONG
$4\pi$
%% WRONG
$6\pi$

%% PROBLEM
Find the centroid of the triangle in the $xy$-plane with vertices
$(0,0)$, $(2,1)$, $(2,0)$.

%% CORRECT
$({4\over3},{1\over3})$
%% WRONG
$({3\over4},{1\over4})$
%% WRONG
$({5\over4},{1\over4})$
%% WRONG
$({5\over4},{1\over3})$
%% WRONG
$({4\over3},{1\over4})$


%% PROBLEM
Compute the volume of the portion of the solid region bewteen the spheres
$\rho = 1$ and $\rho = 2$ that lies inside the upper nappe of the cone $x^2
+y^2 = z^2$.
$$\epsfxsize=1truein\epsfbox{final.2.eps}$$

%% CORRECT
$7\pi(2-\sqrt{2})/3$
%% WRONG
$5\pi(\sqrt{2}-1)/3$
%% WRONG
$14\pi/3$
%% WRONG
$5\sqrt{2}\pi/3$
%% WRONG
$7\pi$


%% PROBLEM
Find the average value of the height function $f(x,y,z) = z$ in
the solid region $D$ inside the cylinder $x^2+y^2 = 1$ between the
planes $x + y + z = 4$ and $z=0$. The volume of $D$ is $4\pi$.
$$\epsfxsize=1truein\epsfbox{final.3.eps}$$

%% CORRECT
$2.06$
%% WRONG
$2.84$
%% WRONG
$2.57$
%% WRONG
$3.21$
%% WRONG
$2.28$

%% PROBLEM
Compute the Jacobian ${\partial(x,y)\over\partial(u,v)}$ of the
change of coordinates $u = 5x + 2y$ and $v = y - 2x$.

%% CORRECT
${1\over9}$
%% WRONG
$9$
%% WRONG
$5$
%% WRONG
${1\over5}$
%% WRONG
$-12$


%% PROBLEM
Let $\C$ be the curve defined by $\r(t) = t^3\i + t^2\j$, $0 \le
t \le 1$. Compute $\int_\C(9x+2\sqrt{y})\,ds$.

%% CORRECT
${13^{3/2}\over6}$
%% WRONG
${17^{3/2}\over2}$
%% WRONG
${15^{3/2}\over3}$
%% WRONG
${11^{3/2}\over2}$
%% WRONG
${9\over2}$


%% PROBLEM
Use the Fundamental Theorem of Line Integrals to compute $\int_\C
(2xy+1)\,dx + (x^2+3y^2-z)\,dy + (6z-y)\,dz$.
where $\C$ is a smooth curve from $(0,0,0)$ to $(1,2,3)$.

%% CORRECT
32
%% WRONG
16
%% WRONG
8
%% WRONG
64
%% WRONG
4


%% PROBLEM
Compute the surface area of the portion of the paraboloid $x^2+y^2+z=9$
above the plane $z=5$.
$$\epsfxsize=1truein\epsfbox{final.4.eps}$$

%% CORRECT
${\pi\over6}(17^{3/2}-1)$
%% WRONG
${\pi\over2}(11^{3/2}-1)$
%% WRONG 
${\pi\over12}(13^{3/2}-1)$
%% WRONG
${\pi\over3}(15^{3/2}-1)$
%% WRONG
${13\pi\over3}$

%% PROBLEM
Let $\Sigma$ be the part of the sphere $x^2+y^2+z^2 = 4$ above the triangle
in the $xy$-plane with vertices $(0,0,0)$, $(1,1,0)$, and $(1,0,0)$. Let
$\F = y\i - x\j + 2x\k$ and let $\n$ be the upward unit normal vector to
$\Sigma$. Compute the flux integral $\iint_\Sigma \F\bdot\n\,d\sigma$.

%% CORRECT
2/3
%% WRONG
3/4
%% WRONG
4/5
%% WRONG
1/2
%% WRONG
7/8


%% PROBLEM
Let $\C$ be the intersection of the plane $y+z=1$ with the cylinder
$x^2+y^2 = 4$, oriented counterclockwise. Let $\F = x^2\i + xy\j +
z^2\k$. Use Stokes' Theorem to compute $\int_\C \F\bdot d\r$.
$$\epsfxsize=2truein\epsfbox{final.5.eps}$$

%% CORRECT
0
%% WRONG
$2\pi$
%% WRONG
$3\pi/4$
%% WRONG
$5\pi/16$
%% WRONG
$\pi/2$


%% PROBLEM
Let $\Sigma$ be the unit sphere with outward unit normal vector $\n$. Let
$\F = x^3\i + y^3\j + z^3\k$. Use the Divergence Theorem to compute the
flux integral $\iint_\Sigma \F\bdot\n\,d\sigma$.

%% CORRECT
$12\pi/5$
%% WRONG
$16\pi/3$
%% WRONG
$6\pi$
%% WRONG
$9\pi/2$
%% WRONG
$2\pi$

%% END
\end
