\input calc
\input epsf
\def\dover{\over\displaystyle}
\def\problemspace{0.2in}

%% TITLE
\noindent
{\bf Math 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\quad{\sl December 14, 1992}\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 6 points each.

\def\answerspace{0.25in}


%% PROBLEM
Use the Divergence Theorem to calculate the flux integral
$\iint_\Sigma \F\cdot\n\,dS$
where $\F = y z^2 \i + z x^2 \j + x y^2 \k$, $\Sigma$ is the
sphere $x^2 + y^2 + z^2 = 9$, and $\n$ is the outward unit normal vector to $\Sigma$.

%% CORRECT
$0$
%% WRONG
$2$
%% WRONG
$3$
%% WRONG
$36\pi$
%% WRONG
$4\pi^2$


%% PROBLEM
Use Stokes' Theorem to calculate the flow integral $\int_\CC \F\cdot d\r$
where $\CC$ is the boundary of the rectangle with vertices
$(0,0,1)$, $(1,0,1)$, $(1,1,0)$, $(0,1,0)$ in the plane $y + z = 1$ and 
$\F = (x + y)\i + (y + z)\j + (x + z)\k$.
$$\epsfxsize=1.5truein\epsfbox{final.1.eps}$$

%% CORRECT
$-2$
%% WRONG
$-1$
%% WRONG
$0$
%% WRONG
$1$
%% WRONG
$2$


%% PROBLEM
 Let $\F = x\i + 3y\j - 2x z\k$
and let $\Sigma$ be the portion of the cylinder $y^2 + z^2 = 1$ above the 
$xy$-plane between $x = 0$ and $x = 1$. Determine which of the following 
integrals gives the value of the flux integral $\iint_\Sigma\F\cdot\n\,d\sigma$
where $\n$ is the upward unit normal vector to $\Sigma$.
$$\epsfxsize=1.5truein\epsfbox{final.2.eps}$$

%% CORRECT
$\int_{-1}^1\int_0^1 3y^2(1-y^2)^{-1/2} - 2x(1-y^2)^{1/2}\,dx\,dy$
%% WRONG
$\int_0^{2\pi}\int_0^1 3r^3(1-r^2)^{1/2}r\,dr\,d\theta$
%% WRONG
$\int_{-1}^1\int_0^1 3x^2(1-y^2)^{1/2} - 2y(1-y^2)^{3/2}\,dx\,dy$
%% WRONG
$\int_0^{2\pi}\int_0^1 3r^2\cos(\theta)(1-r^2)^{3/2} \,dr\,d\theta$
%% WRONG
$\int_0^1\int_0^1 3y(1-y^2)^{1/2} - 2x^2(1-y^2)^{-1/2}\,dx\,dy$

%% PROBLEM
Calculate the surface area of the portion of the paraboloid $x^2 + y + z^2 = 4$ in the first octant.
%% CORRECT
$9.04$
%% WRONG
$10.21$
%% WRONG
$13.37$
%% WRONG
$8.62$
%% WRONG
$5.79$


%% PROBLEM
Let $\CC$ be the curve parameterized by $\r(t) = (t^3 - 1)\i + t^2\j +
t\k$, $0\le t \le 2$. Compute the flow integral\hfil\break
$\int_\CC (y - x)\,dx + (z - y)\,dy + 2xz\,dz$.

%% CORRECT
$1.33$
%% WRONG
$4.5$
%% WRONG
$6.00$
%% WRONG
$8.67$
%% WRONG
$2.50$


%% PROBLEM
Let $\CC$ be the directed line segment from $(-1,2,-1)$ to $(5,0,2)$.
Calculate the line integral $\int_\CC (x + y + z)\,ds$.
%% CORRECT
$24.5$
%% WRONG
$\sqrt{15}/2$
%% WRONG
$\sqrt{31}/2$
%% WRONG
$48.5$
%% WRONG
$72.5$


%% PROBLEM
Let $\F(x,y,z) = {x\dover y}\i + {y\dover z}\j + {z\dover x}\k$. Compute
$\div\F$.
%% CORRECT
$xz + xy + yz\dover xyz$
%% WRONG
${y\dover z^2}\i + {z\dover x^2}\j + {x\dover y^2}\k$
%% WRONG
$x^2y + y^2z + xz^2\dover x^2 y^2 z^2$
%% WRONG
${1\dover y}\i + {1\dover z}\j + {1\dover x}\k$
%% WRONG
${-x\dover y^2} - {y\dover z^2} - {z\dover x^2}$


%% PROBLEM
Determine which of the following vector fields is depicted below.
$$\epsfxsize=1.5truein\epsfbox{final.3.eps}$$

%% CORRECT
$(-y\i + x\j)\dover \sqrt{x^2 + y^2}$
%% WRONG
${x\dover 5}\i + {y\dover 5}j$
%% WRONG
${y\dover 5}\i - {x\dover 5}\j$
%% WRONG
${x\i - y\j}\dover\sqrt{x^2 + y^2}$
%% WRONG
$x^2\i + y^2\j$


%% PROBLEM
Let $R$ be the region bounded by the lines $x + 2y = 3$, $x + 2y = 5$, $2x
- y = 1$, and $2x - y = 2$. Use a substitution to calculate the integral
$\iint_R (x - y)\,dA$.
$$\epsfxsize=1.5truein\epsfbox{final.4.eps}$$

%% CORRECT
$1/25$
%% WRONG
$1/16$
%% WRONG
$1/5$
%% WRONG
$1/4$
%% WRONG
$1/12$


%% PROBLEM
Which of the following integrals gives the volume of the portion of the
solid sphere $x^2 + y^2 + z^2 \le 4$ above the $xy$-plane and below the
cone $x^2 + y^2 = 3z^2$.
$$\epsfxsize=1.5truein\epsfbox{final.5.eps}$$

%% CORRECT
$\int_0^{2\pi}\int_{\pi/3}^{\pi/2}\int_0^2\rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^{\pi/2}\int_0^2\rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^{1}\int_0^{\sqrt{4-r^2}}r\,dz\,dr\,d\theta$
%% WRONG
$\int_0^{\pi/2}\int_0^{1}\int_0^{\sqrt{4-r^2}}r\,dz\,dr\,d\theta$
%% WRONG
$\int_0^{\pi/2}\int_{\pi/6}^{\pi}\int_0^2\rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta$


%% PROBLEM
A mound of earth has the shape of an inverted paraboloid
$z = 1 - x^2 - y^2$. The density of material in the mound is given by 
$\delta(x,y,z) = 1 - z$. Compute the total mass of the mound.
$$\epsfxsize=2.0truein\epsfbox{final.6.eps}$$

%% CORRECT
$\pi/3$
%% WRONG
$\pi/2$
%% WRONG
$\pi/4$
%% WRONG
$\pi/9$
%% WRONG
$\pi/6$


%% PROBLEM
Find the average value of the function $f(x,y) = x^2$ over the triangular
region bounded by the lines $y = x$, $y = 0$ and $x = 2$.

%% CORRECT
$2$
%% WRONG
$4$
%% WRONG
$6$
%% WRONG
$1$
%% WRONG
$3$


%% PROBLEM
Compute the area of the region inside the cardioid
$r = 3 + 2\cos(\theta)$.
$$\epsfxsize=1.5truein\epsfbox{final.7.eps}$$

%% CORRECT
$11\pi$
%% WRONG
$9\pi$
%% WRONG
$6\pi$
%% WRONG
${16\dover 3}\pi$
%% WRONG
${32\dover 3}\pi$


%% PROBLEM
Compute $\int_0^1\int_x^{x^2}\int_0^{y - x} (x + y)\,dz\,dy\,dx$
%% CORRECT
$1/70$
%% WRONG
$1/35$
%% WRONG
$1/15$
%% WRONG
$1/105$
%% WRONG
$1/210$


%% PROBLEM
Rewrite the integral $\int_1^4 \int_{4/x^2}^{(21-5x)/4} f(x,y)\,dy\,dx$
by reversing the order of integration.

%% CORRECT
$\int_{1/4}^4\int_{2/\sqrt{y}}^{(21-4y)/5} f(x,y)\,dx\,dy$
%% WRONG
$\int_{4/x^2}^{(21-5x)/4} \int_1^4 f(x,y)\,dx\,dy$
%% WRONG
$\int_{1}^4\int_{4/y^2}^{(21-4y)/5} f(x,y)\,dx\,dy$
%% WRONG
$\int_{0}^2\int_{2/y}^{(21-5y)/4} f(x,y)\,dx\,dy$
%% WRONG
$\int_{0}^2\int_{2/\sqrt{y}}^{(21-5y)/4} f(x,y)\,dx\,dy$


%% PROBLEM
Find the minimum value of the function $f(x,y) = x - y$ on the ellipse 
$4x^2 + y^2 = 1$.

%% CORRECT
$-1.12$
%% WRONG
$-2.23$
%% WRONG
$-1.00$
%% WRONG
$-0.50$
%% WRONG
$-1.73$


%% PROBLEM
Which of the following statements applies to the function
$f(x,y) = x^3 - 3x^2 + 3y^2$.

%% WRONG
$(0,0)$ is {\sl not} a critical point of $f$
%% WRONG
$f$ has a saddle point at $(2,0)$
%% WRONG
$f$ has a local maximum at $(0,0)$
%% CORRECT
$f$ has a local minimum at $(2,0)$
%% WRONG
{\sl none of the above}
%% STET

%% PROBLEM
Find all of the critical points of the function
$f(x,y) = x^2(2 - 3x^2) + 12xy(x^2-1)$.

%% CORRECT
$(1,1/3)$, $(-1,-1/3)$, $(0,0)$
%% WRONG
$(1,1/3)$, $(1,-1/3)$, $(-1,1/3)$, $(-1,-1/3)$
%% WRONG
$(1,-1/3)$, $(-1,1/3)$, $(0,0)$
%% WRONG
$(1,1/3)$, $(-1,1/3)$, $(-1,-1/3)$, $(0,0)$
%% WRONG
$(1,1/3)$, $(1,-1/3)$, $(0,0)$


%% PROBLEM
Find the maximum value of $f(x,y) = e^{xy - x - 2y}$ on the region
$x + y \le 7$, $x \ge 0$, $y \ge 0$.

%% CORRECT
$e^2$
%% WRONG
$e$
%% WRONG
$e^{-1}$
%% WRONG
$1$
%% WRONG
$e^4$


%% PROBLEM
Determine the equation of the plane tangent to the surface $x^2 - y z = 4$
at the point $(2,0,0)$.

%% CORRECT
$x = 2$
%% WRONG
$2x - y - z = 0$
%% WRONG
$2x - y - z = 4$
%% WRONG
$4x - y = 0$
%% WRONG
$4x - y = 8$


%% PROBLEM
Find the direction in which the function $f(x,y,z) = x^3y - y^2z + z^4$
increases most rapidly at the point $(1,2,-1)$.

%% CORRECT
$6\i + 5\j - 8\k$
%% WRONG
$3\i - 2\j + 4\k$
%% WRONG
$3\i - \j + 3\k$
%% WRONG
$\k$
%% WRONG
$6\i + 4\j - 4\k$


%% PROBLEM
Let $f(x,y) = x^3 y^2$. If $x = r\cos(\theta)$ and $y = r\sin(\theta)$,
determine which of the expressions below gives $\d{f}/d\theta$.

%% CORRECT
$r x^2 y(2x\cos(\theta) - 3 y \sin(\theta))$
%% WRONG
$r x^2 y(3y\cos(\theta) + 2 x \sin(\theta))$
%% WRONG
$-3r^5\cos^2(\theta)\sin^3(\theta)$
%% WRONG
$2r^5\cos^4(\theta)\sin(\theta)$
%% WRONG
$6rx^2 y(\sin(\theta) - \cos(\theta)$


%% PROBLEM
Determine the equation of the line perpendicular to the plane
$2x - 3y + 4z = 5$ through the point $(1,0,0)$.

%% CORRECT
$x = 1 + 2t,\quad y = -3t,\quad z = 4t$
%% WRONG
$x = 2 + t,\quad y = -3+t,\quad z = 4+t$
%% WRONG
$x = 1 + t,\quad y = t,\quad z = t$
%% WRONG
$x = 1 + t/2,\quad y = -t/3,\quad z = t/4$
%% WRONG
$x = 5 - 4t,\quad y = -3t + 3,\quad z = 4t - 4$


%% PROBLEM
Compute the angle in radians between the lines \hbox{$x = 2 - t$},
\hbox{$y = 3 - 2t$}, \hbox{$z = 4 - 3t$},\hfil\break and \hbox{$x = 1 + t$},
\hbox{$y = 4 - t$}, \hbox{$z = 4t$}.

%% CORRECT
$2.34$
%% WRONG
$0.79$
%% WRONG
$1.05$
%% WRONG
$1.67$
%% WRONG
$2.09$


%% PROBLEM
A charged particle with mass $m = 2\times10^{-2}$ moves under the 
influence of a changing electrical force,\hfil\break $\F(t) = (1-t)\i + t^2\j$
($t$ in seconds). If the particle is initially at rest at the origin, 
determine the particle's position after 2 seconds.

%% CORRECT
$(100\i + 200\j)/3$
%% WRONG
$(2\i + 4\j)/3$
%% WRONG
$(50\i + 100\j)/3$
%% WRONG
$(\i + 8\j)/6$
%% WRONG
$(50\i + 400\j)/6$



%% END
\end
