\magnification=\magstep1
\input epsf
\input calc
\def\problemspace{0.25in}

%% TITLE
{\bf Math 225: Calculus III}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Final Exam}\quad{\sl May 8, 1996}\hfill Section:\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 5 points each. You start
with 25 points.

\def\answerspace{0.25in}

%1
%% PROBLEM
Let $\a = -2\i  + 4\j + 5\k$ and $\b = \i + 3\j - 2\k$. Compute $\a \cdot\b$.

%% CORRECT
$0$
%% WRONG
$5$
%% WRONG
$-2$
%% WRONG
$-23\i + \j + 10\k$
%% WRONG
$-2\i + 12\j - 10\k$


%2
%% PROBLEM
Let $\a = \i - 3\k$ and $\b = 2\i - \j + \k$. Compute $\a \times \b$.
%% CORRECT
$-3\i - 7\j - \k$
%% WRONG
$-3\i - \j + 5\k$
%% WRONG
$5$
%% WRONG
$5\k$
%% WRONG
$-1$


%3
%% PROBLEM
Determine the parametric equations of the line through the point
$(1,-2,1)$ parallel to the vector $\a = 4\i + \j - 3\k$.

%% CORRECT
\vtop{\hsize = 1.2truein$\eqalign{x &= 1 + 4t\cr y &= -2 + t\cr z &= 1 - 3t}$}
%% WRONG
\vtop{\hsize = 1.2truein$\eqalign{x &= 4 + t\cr y &= 1 - 2t\cr z &= -3 + t}$}
%% WRONG
\vtop{\hsize = 1.2truein$\eqalign{x &= 4t\cr y &= t\cr z &=  -3t}$}
%% WRONG
\vtop{\hsize = 1.2truein$\eqalign{x &= 1 + t\cr y &= -2 + t\cr z &= t}$}
%% WRONG
\vtop{\hsize = 1.2truein$\eqalign{x &= 1 - 4t\cr y &= 2 - t\cr z &= 3t}$}

%4
%% PROBLEM
Determine the equation of the plane perpendicular to the vector $\a =
\i + 3\j - 5\k$ passing through the point $(4, -1,2)$.

%% CORRECT
$x + 3y - 5z = -9$
%% WRONG
$4x - y + 2z = 0$
%% WRONG
$4x - y + 2z = -9$
%% WRONG
$(x+4) + 3(y+1) - 5(z+2) = 0$
%% WRONG
$x + 3y - 5z = 0$



%5
%% PROBLEM
Determine which of the following integrals gives the length of the
curve parameterized by\hfil\break
$\r(t) = e^{-t}\cos(t)\i + e^{-t}\sin(t)\j - te^{-t}\k$, $0 \le t \le 1$.

%% CORRECT
$\int_0^{1} e^{-t}\sqrt{2 + (1-t)^2}\, dt$
%% WRONG
$\int_0^{1}\sqrt{e^{-2t} + 1}\, dt$
%% WRONG
$\int_0^{1} \sqrt{e^{-t} + (1-t)^2}\, dt$
%% WRONG
$\int_0^{1} e^{-t}\sqrt{(1+\cos(t))^2 + (1+\sin(t))^2 + 1}\, dt$
%% WRONG
$\int_0^{1} e^{-t}\sqrt{2\cos(t)\sin(t) + (1-t)^2}\, dt$


%6
%% PROBLEM
Find the parametric equations of the line tangent to the curve
$\r(t) = \log(1/t)\i + t\j + t^2\k$ at the point $(0,1,1)$.

%% CORRECT
\vtop{\hsize = 1.2in$\eqalign{x &= -t\cr y &= 1 + t\cr z &= 1 + 2t}$}
%% WRONG
\vtop{\hsize = 1.2in$\eqalign{x &=1 - 1/t\cr y &= 3\cr z &=1 + 2t}$}
%% WRONG
\vtop{\hsize = 1.2in$\eqalign{x &= -1\cr y &= 1 + t\cr z &= 1 + 2t^2}$}
%% WRONG
\vtop{\hsize = 1.2in$\eqalign{x &=  - 1/t\cr y &= 1+t\cr z &= 1 + 2t^2}$}
%% WRONG
\vtop{\hsize = 1.25truein$\eqalign{x &= - t\cr y &= t\cr z &= 2t}$}


%7
%% PROBLEM
Let $f(x,y) = x^3e^{y^2} + \sin(xy)$. Compute $f_{xy}$.
%% CORRECT
$6x^2 y e^{y^2} + \cos(xy) - x y \sin(xy)$
%% WRONG
$(3x^2+ 4x^3y^2)e^{y^2} - x^2\sin(xy)$
%% WRONG
$(3x^2 + 2x^3y)e^{y^2} + (x + y)\cos(xy)$
%% WRONG
$(3x^2e^{y^2} + y\cos(xy))\i+ (2x^3ye^{y^2} + x\cos(xy))\j$
%% WRONG
$6x^2ye^{y^2} - xy\cos(xy)$

%8
%% PROBLEM
Let $f(x,y) = x\sqrt{x^2 +  y^2}$ and suppose $x$ and $y$ are functions of
$u$ and $v$. If $\d{x}/du = 3$, $\d{x}/dv = -2$, $\d{y}/du = -4$ and
$\d{y}/dv = 7$ when $(x,y) = (1,2)$, compute $\d{f}/du$ at that point.
%% CORRECT
$2\sqrt{5}$
%% WRONG
$4/\sqrt{5}$
%% WRONG
$-1/\sqrt{5}$
%% WRONG
$4\sqrt{5}$
%% WRONG
$2/\sqrt{5}$


%9
%% PROBLEM
Calculate the derivative of the function $f(x,y) = x^3y - y^2$ in the
direction $30^\circ$ above horizontal at the point $(1,-1)$.
%% CORRECT
$-1.098$
%% WRONG
$0$
%% WRONG
$-2.130$
%% WRONG
$-1.703$
%% WRONG
$-2.517$


%10
%% PROBLEM
Determine the equation of the plane that is tangent to the graph of\hfil\break
$f(x,y) = x^3 + y^3$ at the point $(-1,2,9)$.
%% CORRECT
$3x + 12y - z = 12$
%% WRONG
$3x + 12y = 21$
%% WRONG
$3x + 12y - z = 0$
%% WRONG
$3x^3 + 3y^3 + z = 0$
%% WRONG
$3x^2(x+1) + 3y^2(y-2) + (z-9) = 0$


%11
%% PROBLEM
Determine which of the following statements describes the graph of $f(x,y)
= 4x^3 - 6x^2y + 3 y^2$ over the point $(1,1)$.

%% CORRECT
{\sl a saddle point}
%% WRONG
{\sl a local maximum}
%% WRONG
{\sl a local minimum}
%% WRONG
{\sl not a critical point}
%% WRONG
{\sl not continuous}

%12
%% PROBLEM
Find the maximum of $f(x,y) = x^4 y$ subject to the constraint $x^2 + 5y^2 = 5$.

%% CORRECT
$7.155$
%% WRONG
$10.062$
%% WRONG
$11.180$
%% WRONG
$6.708$
%% WRONG
$8.944$


%13
%% PROBLEM
Reverse the order of integration in the double integral
$\int_{-3}^3\int_1^{10-x^2} f(x,y)\,dy\,dx$.
%% CORRECT
$\int_{1}^{10}\int_{-\sqrt{10-y}}^{\sqrt{10-y}} f(x,y)\,dx\,dy$
%% WRONG
$\int_1^{10-x^2}\int_{-3}^3 f(x,y)\,dx\,dy$.
%% WRONG
$\int_{-3}^3\int_1^{10-y^2} f(x,y)\,dx\,dy$.
%% WRONG
$\int_{1}^{10}\int_{0}^{\sqrt{10-y}} f(x,y)\,dx\,dy$
%% WRONG
$\int_{0}^{10}\int_{-\sqrt{10-y}}^{\sqrt{10}} f(x,y)\,dx\,dy$

%14
%% PROBLEM
Calculate the area of the region bounded above by the circle of radius
$1$ and below by the portion of the spiral $r = \theta/\pi$ above the
$x$-axis in polar coordinates.

\centerline{\epsfxsize=2.5truein\epsfbox{final.14.eps}}

%% CORRECT
${1\over3}\pi$
%% WRONG
${1\over2}\pi - {1\over6}\pi^3$
%% WRONG
${1\over6}\pi^2$
%% WRONG
${1\over2}\pi$
%% WRONG
${1\over3}\pi^2 - {1\over 6}\pi^3$



%15
%% PROBLEM
Which of the following integrals gives the volume of the solid bounded
by the $xy$-plane, the $yz$-plane, the parabolic sheet $x + y^2 = 1$, and the
cylinder $y^2 + z^2 = 1$.

\centerline{\epsfxsize=3.5truein\epsfbox{final.15.eps}}

%% CORRECT
$\int_0^1\int_{-\sqrt{1-x}}^{\sqrt{1-x}}\int_0^{\sqrt{1 - y^2}} 1\,dz\,dy\,dx$
%% WRONG
$\int_0^1\int_0^{1-y^2}\int_0^{\sqrt{1-y^2}} 1\,dz\,dx\,dy$
%% WRONG
$\int_{-1}^1\int_{1-y^2}^{1}\int_0^{\sqrt{1-y^2}} 1\,dz\,dx\,dx$
%% WRONG
$\int_0^1\int_{y^2-1}^{1-y^2}\int_0^{x+y^2} 1\,dz\,dx\,dy$
%% WRONG
$\int_{-1}^{1}\int_0^{x+y^2}\int_0^{y^2 + z^2} 1\,dz\,dy\,dx$



%16
%% PROBLEM
Let $D$ be the part of the cylinder $x^2 + y^2 = 4$ below the plane $y
+ z = 2$ and above the $xy$-plane. The volume of $D$ is $8\pi$.
Compute the $z$-coordinate of the centroid of $D$.

\centerline{\epsfxsize=2.5truein\epsfbox{final.16.eps}}

%% CORRECT
$1.25$
%% WRONG
$1.33$
%% WRONG
$1.50$
%% WRONG
$0.67$
%% WRONG
$1.75$


%17
%% PROBLEM
Determine which of the following integrals gives the volume of the part
of the solid\hfil\break
sphere $x^2 + y^2 + z^2 \le 1$ cut out by the the upper nappe of
the cone $3z^2 = x^2 + y^2$.

\centerline{\epsfxsize=2.5truein\epsfbox{final.17.eps}}

%% CORRECT
$\int_0^{2\pi}\int_0^{\pi/3}\int_0^{1} \rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^{\pi/6}\int_0^{1} \rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^{\sin(\phi)}\int_0^{\rho^2} 1\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^{\pi/4}\int_0^{1} \rho\sin(\theta)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^{\pi/2}\int_0^{1} \rho^2\cos(\phi)\,d\rho\,d\phi\,d\theta$



%18
%% PROBLEM
Consider the change of variables $u = \log(x^2+y^2)$, $v = y/x$.
Compute the Jacobian determinant ${\partial(x,y)\over\partial(u,v)}$.

%% CORRECT
${e^u\over2(1+v^2)}$
%% WRONG
$2e^{-u}(1-v^2)$
%% WRONG
$e^{-u}(1+v^2)$
%% WRONG
${e^u\over(1-v^2)}$
%% WRONG
${2(x^2-y^2)\over x^2+y^2}$

%19
%% PROBLEM
Compute the line integral $\int_\CC z\,dx + (x-1)z\,dy - z(x+y))\,dz$
where $\CC$ is the curve parameterized by $\r(t) = e^t\i - e^{-t}\j + t\k$,
$0\le t \le 2$.

%% CORRECT
$2$
%% WRONG
$0$
%% WRONG
$2e^2 - e^{-2}$
%% WRONG
$(e^2 - e^{-2})/2$
%% WRONG
$(3e^2 - e^{-2})/2$

%20
%% PROBLEM
Use the Fundamental Theorem of Line Integrals to calculate
$$\int_\CC
(x^2+y^2+z^2)\,dx + 2y(x - z^3)\,dy + z(2x + 3z - 3y^2z)\,dz
$$
where $\CC$ is the curve parameterized by
$$\r(t) = \cos[2\pi(1+t^2)]\i + (1-t^2)\j + t\k, \qquad 0\le t \le 1$$

%% CORRECT
$1$
%% WRONG
$-1$
%% WRONG
$0$
%% WRONG
$4/3$
%% WRONG
$-7/3$


%21
%% PROBLEM
Let $\CC$ be the boundary of the triangle with vertices
$$(0,0),\qquad (1,0), \qquad (0,2)$$
oriented counter-clockwise. Use Green's Theorem to compute
the line integral
$$\int_\CC (x^3 + x y^2)\,dx + (y^3 + x^2 y)\,dy$$

%% CORRECT
$0$
%% WRONG
$2/3$
%% WRONG
$-1$
%% WRONG
$10/3$
%% WRONG
$1$

%22
%% PROBLEM
Calculate the surface area of the part of the paraboloid $x^2 + y^2 + z
= 9$ above the $xy$-plane.

%% CORRECT
${\pi\over 6}[37^{3/2}-1]$
%% WRONG
$9\sqrt{9 - \pi^2}$
%% WRONG
${\pi\over 3}[(10)^{3/2}-1]$
%% WRONG
${\pi\sqrt{3}\over 2}$
%% WRONG
$3\sqrt{1 + 3\pi^2}$


%23
%% PROBLEM
Let $\Sigma$ be the portion of the sphere of radius 2 in the first octant
and let $\n$ be the unit upward normal vector to $\Sigma$. Compute the flux
integral $\iint_\Sigma (xz\i + y^2\k)\cdot\n\,d\sigma$.

\centerline{\epsfxsize=1.5truein\epsfbox{final.23.eps}}

%% CORRECT
$2\pi$
%% WRONG
$\pi$
%% WRONG
0
%% WRONG
${3\pi\over2}$
%% WRONG
${\pi\over2}$

%24
%% PROBLEM
Let $\CC$ be the boundary of the triangle cut out from the plane $x + y +
z = 1$ by the first octant. Use Stokes' Theorem to calculate the line
integral $\int_\CC (z\i + x\j + y\k)\cdot d\r$

\centerline{\epsfxsize=1.5truein\epsfbox{final.24.eps}}

%% CORRECT
${3\over2}$
%% WRONG
$3$
%% WRONG
${1\over2}$
%% WRONG
$0$
%% WRONG
$2$

%25
%% PROBLEM
Let $D$ be the solid rectangular box defined by $0 \le x \le 3$,
$0\le y \le 2$, and $0 \le z \le 1$. Let $\Sigma$ be the boundary of
this box and let $\n$ be its outward unit normal vector. Use the
Divergence Theorem to express the flux integral
$$\iint_\Sigma[(x^2y+z)\i + (y^2z-x)\j +  (x z^2+y)\k]\cdot\n\,d\sigma$$
as a triple integral.

%% CORRECT
$2\int_0^1\int_{0}^2\int_{0}^3 (xy + yz + xz)\,dx\,dy\,dz$
%% WRONG
$\int_0^1\int_{0}^2\int_{0}^3 0\,dx\,dy\,dz$
%% WRONG
$\int_0^1\int_{0}^2\int_{0}^3 (x^2y + y^2z + xz^2-x+y+z)\,dx\,dy\,dz$
%% WRONG
$\int_0^1\int_{0}^2\int_{0}^3 {x\i + y\j + z\k\over xz}\,dx\,dy\,dz$
%% WRONG
$2\int_0^1\int_{0}^2\int_{0}^3 (x\i + y\j + z\k)\,dx\,dy\,dz$

%% END
\end

%% END
\bye
