\input calc
\def\ovr{\over\displaystyle}
\def\problemspace{.25in}

%% TITLE
{\bf Math 225: Calculus III}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Final Exam}\quad{\sl December 19, 1990}\hfill Score:\hbox to 2.5truein{\hrulefill}

\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 25 multiple
choice questions worth 6 points each.

\bigskip

%1
%% PROBLEM
A vector $\a$ is perpendicular to a vector $\b$ if
%% CORRECT
$\a\bdot\b = 0$
%% WRONG
$\a\times\b = 0$
%% WRONG
$\a + \b = 0$
%% WRONG
$\a = c\b$, some scalar $c$
%% WRONG
$\a\bdot(\b\times\a) = 0$

%2
%% PROBLEM
Compute the projection of $\a = 2\i + 3\j + 5\k$ onto $\b = \i + 2\j + 2\k$.
%% CORRECT
$2\i + 4\j + 4\k$
%% WRONG
$2\i + 3\j + 5\k$
%% WRONG
$2\i + 6\j + 10\k$
%% WRONG
$\i + {3\ovr2}\j + {5\ovr2}\k$
%% WRONG
${1\ovr2}\i + \j + \k$

%3
%% PROBLEM
Let $\a = \i + \j + \k$ and $\b = 2\i + 3\j + 4\k$. Compute $\a\times\b$
%% CORRECT
$\i - 2\j + \k$
%% WRONG
$2\i + 3\j + 4\k$
%% WRONG
$\i + 2\j + 3\k$
%% WRONG
$-\i + 2\j - 3\k$
%% WRONG
$\0$

%4
%% PROBLEM
Determine the symmetric equations of the line through the points $(-1,0,3)$ and
$(2,1,-1)$.
%% CORRECT
${x+1\ovr3} = y = {z-3\ovr-4}$
%% WRONG
${x+1\ovr3} = y,\quad z = 3$
%% WRONG
$x = -1,\quad y = {z+1\ovr-4}$
%% WRONG
${x-2\ovr-1}= {z+1\ovr3},\quad y = 1$
%% WRONG
${x-2\ovr-1} = y-1 = {z+1\ovr3}$

%5
%% PROBLEM
Determine the equation of the plane containing the points $(0,0,0)$,
$(1,2,3)$ and $(4,5,6)$.
%% CORRECT
$x - 2y + z = 0$
%% WRONG
$2x - y = 0$
%% WRONG
$5x - 4y = 0$
%% WRONG
$4x + y - 2z = 0$
%% WRONG
$4x - 2y - z = 0$

%6
%% PROBLEM
Find $\lim_{(x,y)\to(0,0)}{xy\ovr x^2 + y^2}$.
%% CORRECT
{\sl Does not exist}
%% WRONG
$1$
%% WRONG
$1\ovr2$
%% WRONG
$0$
%% WRONG
$-1$

%7
%% PROBLEM
Suppose the motion of a particle is described by
$\r(t) = e^t\i + e^{-t}\j + {1\ovr\sqrt{2}}t^2\k$, $t \ge 0$.
Find the particle's speed at time $t = 1$.
%% CORRECT
$3.086$
%% WRONG
$4.500$
%% WRONG
$3.765$
%% WRONG
$2.021$
%% WRONG
$2.920$

%8
%% PROBLEM
Suppose a particle moves along the path
$\r(t) = \cos(t^2)\i + \sin(t^2)\j + t^2\k$, $0 \le t \le \sqrt{2\pi}$.
Find the total distance travelled by the particle.
%% CORRECT
$8.886$
%% WRONG
$6.283$
%% WRONG
$4.443$
%% WRONG
$2.507$
%% WRONG
$12.566$

%9
%% PROBLEM
A 1~meter cube is being compressed. After 2 seconds, its height is 99~cm and
is decreasing at a rate of $1 {\rm cm \ovr \rm sec}$ while its depth and width
are 101~cm and are increasing at a rate of $0.5 {\rm cm\ovr \rm sec}$. Find
the rate of change in the volume of the cube after 2~seconds of being compressed.
%% CORRECT
$-202 {\rm cm^3 \ovr \rm sec}$
%% WRONG
$0 {\rm cm^3 \ovr \rm sec}$
%% WRONG
$-99 {\rm cm^3 \ovr \rm sec}$
%% WRONG
$-9801 {\rm cm^3 \ovr \rm sec}$
%% WRONG
$-5 {\rm cm^3 \ovr \rm sec}$

%10
%% PROBLEM
Let $f(x,y,z) = x^2y + y^2z + z^2x$. Find the derivative of $f$ in the
direction of the vector $\sqrt{2}\i + \j + \k$ at the point $(0,1,2)$.
%% CORRECT
${5\over2} + 2\sqrt{2}$
%% WRONG
$4\i + 4\j + \k$
%% WRONG
${3\ovr2} + \sqrt{2}$
%% WRONG
$6 + {\sqrt{2}\ovr2}$
%% WRONG
${\sqrt{2}\ovr2}\i + {1\ovr2}\j + {1\ovr2}\k$

%11
%% PROBLEM
Let $f(x,y,z) = xyz^2 + 2x^2y$. Find the direction of maximum increase in $f$
at the point $(-1,-2,1)$.
%% CORRECT
$6\i + \j + 4\k$
%% WRONG
$-\i -2\j -\k$
%% WRONG
$2\i - \j - \k$
%% WRONG
$4\i - 2\j - \k$
%% WRONG
$-4\i + \j + 2\k$

%12
%% PROBLEM
Let $f(x,y) = x^3 + xy + y^2$. Determine which of the following statements is
true.
%% CORRECT
$f$ has a relative minimum at $({1\ovr6},-{1\ovr12})$.
%% WRONG
$f$ has a relative maximum at $({1\ovr6},-{1\ovr12})$.
%% WRONG
$f$ has a saddle point at $({1\ovr6},-{1\ovr12})$.
%% WRONG
$({1\ovr6},-{1\ovr12})$ is {\bf not} a critical point of $f$
%% WRONG
{\sl None of the above}

%13
%% PROBLEM
Find the minimum of $f(x,y) = x(1 - e^y)$ on the unit square $0\le x \le 1$,
$0\le y \le 1$.
%% CORRECT
$-1.718$
%% WRONG
$0$
%% WRONG
$-1$
%% WRONG
$-0.718$
%% WRONG
$-2.718$

%14
%% PROBLEM
Find the maximum of $f(x,y) = 2y^3 + 6x^2y$ subject to the constraint
$x^2 + y^2 = 4$.
%% CORRECT
$16\sqrt{2}$
%% WRONG
$18$
%% WRONG
$18\sqrt{3}$
%% WRONG
$16$
%% WRONG
$24$

%15
%% PROBLEM
Evaluate $\int_0^2\int_1^{1-x^2} xy\dydx$.
%% CORRECT
${4\ovr 3}$
%% WRONG
${1\ovr 3}$
%% WRONG
$0$
%% WRONG
$1$
%% WRONG
${2\ovr 3}$

%16
%% PROBLEM
Reverse the order of integration in the iterated integral
$\int_{-1}^1\int_{|x|}^1\cos(y^{3/2})\dydx$.
%% CORRECT
$\int_0^1\int_{-y}^{y}\cos(y^{3/2})\dxdy$
%% WRONG
$\int_{|x|}^1\int_{-1}^{1}\cos(y^{3/2})\dxdy$
%% WRONG
$\int_{-1}^1\int_{0}^{|y|}\cos(y^{3/2})\dxdy$
%% WRONG
$\int_0^1\int_0^{y}\cos(y^{3/2})\dxdy$
%% WRONG
$\int_{-1}^1\int_{-y}^{0}\cos(y^{3/2})\dxdy$

%17
%% PROBLEM
Let $\F = xy^3\i + yz^3\j + zx^3\k$. Compute $\curl\F$.
%% CORRECT
$-3yz^2\i - 3x^2z\j -3xy^2\k$
%% WRONG
$x^3 + y^3 + z^3$
%% WRONG
$x^3\i - y^3\j + z^3\k$
%% WRONG
$-3xy^2\i + 3yz^2\j - 3zx^2\k$
%% WRONG
$0$

%18
%% PROBLEM
Which of the following integrals gives the volume of the solid bounded by the
planes $x + y - z = 0$, $x - y + z = 0$, $x = 0$, $y = 0$, and $x + y = 2$.
%% CORRECT
$\int_0^2\int_0^{2-x}\int_{y-x}^{y+x}1\,dz\,dy\,dx$
%% WRONG
$\int_0^2\int_0^{2-x}\int_0^{2-y-x}1\,dz\,dy\,dx$
%% WRONG
$\int_0^2\int_{y-2}^{2-y}\int_{y-x}^{y+x}1\,dz\,dx\,dy$
%% WRONG
$\int_0^2\int_0^{2-y}\int_0^{y+x}1\,dz\,dy\,dx$
%% WRONG
$\int_0^2\int_{2-x}^2\int_{y-x}^21\,dz\,dy\,dx$

%19
%% PROBLEM
Let $D$ be the solid in the first octant that lies below the plane
$z = 1$ and is bounded by $x^2 + y^2 = 9$. Write
$\iiint_D xyz\,dV$ in cylindrical coordinates.
%% CORRECT
$\int_0^{\pi/2}\int_0^3\int_0^1 zr^3\cos(\theta)\sin(\theta)\,dz\drdw$
%% WRONG
$\int_0^3\int_0^{\pi}\int_0^1 zr^2\cos(\theta)\sin(\theta)\,dz\drdw$
%% WRONG
$\int_0^{2\pi}\int_0^1\int_0^3 zr^2\cos(\theta)\sin(\theta)\,dz\drdw$
%% WRONG
$\int_0^{\pi/2}\int_0^3\int_0^1 zr^3\cos(\theta)\sin^2(\theta)\,dz\drdw$
%% WRONG
$\int_0^{\pi/2}\int_0^1\int_0^3 zr^2\cos(\theta)\sin(\theta)\,dz\drdw$

%20
%% PROBLEM
Compute $\iiint_D (x^2 + y^2)\,dV$ where $D$ is the unit sphere $x^2 + y^2 +
z^2 = 1$.
%% CORRECT
$8\pi\ovr15$
%% WRONG
$\pi^2\ovr4$
%% WRONG
$3\pi\ovr5$
%% WRONG
$4\pi^2\ovr3$
%% WRONG
$4\pi\ovr5$

%21
%% PROBLEM
Let $C$ be the curve $\r(t) = t\i + t^2\j$, $0\le t \le 1$. Compute $\int_C
x^3\,dx + xy\,dy$.
%% CORRECT
$13\ovr20$
%% WRONG
$1\ovr2$
%% WRONG
$7\ovr12$
%% WRONG
$3\ovr2$
%% WRONG
$14\ovr15$

%22
%% PROBLEM
Let $\Sigma$ be the portion of the plane $3x + 2y + z = 0$ under the
rectangle $0 \le x \le 2$, $0 \le y \le 3$ in the $xy$-plane. Compute
$\iint_\Sigma z^2 - 12xy \, dS$.
%% CORRECT
$144\sqrt{14}$
%% WRONG
$144$
%% WRONG
$288\sqrt{2}$
%% WRONG
$288\sqrt{7}$
%% WRONG
$288$

%23
%% PROBLEM
Let $\F = xe^{xy}\i - ye^{xy}\j + z\k$, let $\Sigma$ be the sphere
$x^2 + y^2 + z^2 = 4$, and let $\n$ be the outward normal vector to $\Sigma$.
Evaluate the flux integral
$\iint_\Sigma\F\bdot\n\,dS$.
%% CORRECT
$32\pi\ovr3$
%% WRONG
$4\pi\ovr3$
%% WRONG
$8\pi\ovr3$
%% WRONG
$16\pi\ovr3$
%% WRONG
$0$

%24
%% PROBLEM
Let $C$ be the intersection of the paraboloid $x^2 + y^2 + z = 2$ and the
plane $x + y + z = 1$. Compute $\int_C x^2\,dx + y^2\,dy + z^2\,dz$.
%% CORRECT
$0$
%% WRONG
$\pi\ovr3$
%% WRONG
$2\pi\ovr3$
%% WRONG
$4\pi\ovr3$
%% WRONG
$8\pi\ovr3$

%% PROBLEM
Let $\F = -z^2\i + e^z\j + (e^zy - 2xz)\k$ and let $\r(t) = t\i + t^2\j +
t^3\k$, $0 \le t \le 1$.
Evaluate $\int_C \F\bdot d\r$.
%% CORRECT
$e-1$
%% WRONG
$1$
%% WRONG
$0$
%% WRONG
$e - e^{-1}$
%% WRONG
$2e-1$

%% END
\bye
