\vsize=8.5truein

\input epsf
\def\a{{\bf a}}
\def\r{{\bf r}}
\def\F{{\bf F}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\n{{\bf n}}
\def\v{{\bf v}}
\def\w{{\bf w}}
\def\R{{\bf R}}
\def\0{{\bf 0}}
\def\C{{\cal C}}

\def\grad{{\rm grad\,}} \def\curl{{\rm curl\,}} \def\div{{\rm div\,}}
\def\iint{\int\!\!\!\int} \def\iiint{\int\!\!\!\int\!\!\!\int}

\def\problemspace{0.1in}

%% TITLE
\noindent
{\bf MATH 225: Calculus III}\hfill
{\bf Practice Final Exam}\quad{\sl December 2002}\hfill

\def\answersep{\ }

%1
%% PROBLEM
Find the arc length of the curve described by the vector function $\r(t)
= 2\cos(\pi t^2) \i + \sqrt{2} \sin(\pi t^2) \j - \sqrt{2} \sin(\pi t^2)
\k$ for $0 \le t \le 1$.

%% CORRECT
$2\pi$
%% WRONG
$4\pi$
%% WRONG
$2$
%% WRONG
$4$
%% WRONG
$16\pi^2/3$

%2
%% PROBLEM
Let $\v = \langle t,t^2,t^3\rangle$ for a real number $t$, and let $\w =
\langle 1,1,1\rangle$.  Find the vector projection ${\rm proj}_\w(\v)$ of $\v$
on $\w$.

%% CORRECT
${{t+t^2 + t^3}\over{3}}\langle 1,1,1\rangle$
%% WRONG
${{t^2+t^4 + t^6}\over{3}}$
%% WRONG
${{t+t^2 + t^3}\over{\sqrt{3}}}$
%% WRONG
$(t+t^2+t^3)\langle 1,1,1\rangle$
%% WRONG
${{t+t^2 + t^3}\over{\sqrt{3}}}\langle t,t^2,t^3\rangle$

%3
%% PROBLEM
Suppose that the acceleration of an object at time $t \ge 0$ is given
by $\a(t) = 2t \i  + 2\k$, that the object's position at time $t=0$
is $\r(0) = \i + \j$, and that its position at time $t=1$ is $\r(1) =
\0$. Find $\v(0)$, the object's velocity at time $t=0$.

%% CORRECT
$-4/3\i-\j-\k$
%% WRONG
$0$
%% WRONG
$-2\i-\j-\k$
%% WRONG
$t^2\i+2t\k$
%% WRONG
$\i+2\k$

%4
%% PROBLEM
Determine which of the following represents the area of the parallelogram
consisting of the points $$(u+1,2u+3v+1,-u + v+1), \quad 0 \le u \le 1,
\quad 0 \le v \le 1$$

%% CORRECT
$|(\i + 2\j -\k) \times (3\j + \k)|$
%% WRONG
$|(\i + 5\j) \times (\i + \j + \k)|$
%% WRONG
$|(2\i + 3\j) \times (\i + 4\j + 2\k)|$
%% WRONG
$|(\i + 2\j - \k) \times (\i + 4\j + 2\k)|$
%% WRONG
$(\i + 2\j -\k) \cdot (3\j + \k)$

%5
%% PROBLEM
Find equations for the line of intersection of the planes $x-y+z = 2$
and $2x + z = 1$.

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

%6
%% PROBLEM
Compute $\lim_{(x, y) \rightarrow (0,0)} {{x^{2} + y^{4}}\over{\sqrt{x^{2}
+ y^{4}+4} - 2}}$.

%% CORRECT
4
%% WRONG
0
%% WRONG
1
%% WRONG
$-1/2$
%% WRONG
{\sl does not exist}

%7
%% PROBLEM
Let $z$ be the function of $x$ and $y$ defined by equation $xz + e^{z}y =
2$. Find ${{\partial z}\over{\partial y}}$ at the point $(1, 2, 0)$.

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

%8
%% PROBLEM
Find the maximum rate of change of the function $f(x, y, z) = x^{2}y +
e^{x} z^{2}$ at point $(0, 1, -1)$.

%% CORRECT
$\sqrt{5}$
%% WRONG
1
%% WRONG
$\sqrt{2}$
%% WRONG
0
%% WRONG
$\sqrt{3}$

%9
%% PROBLEM
Determine which of the following statements describes the function $$f(x,
y) = x^{3} - y^{3} -2xy + 6$$ at point $(-2/3, 2/3)$.

%% CORRECT
$f$ has a local maximum.
%% WRONG
$f$ has a local minimum.
%% WRONG
$f$ has a saddle point.
%% WRONG
The point is not a critical point.
%% WRONG
The second derivative test is inconclusive.

%10
%% PROBLEM
Find the minimum of $f(x,y,z) = (x-1)^{2} + y^{2} + z^{2}$ subject to
the constraint $x^{2}- yz = 0$.

%% CORRECT
$2/3$
%% WRONG
$1/3$
%% WRONG
$1/2$
%% WRONG
$1$
%% WRONG
$2$

%11
%% PROBLEM
Evaluate $\int_0^1\int_{e^y}^e {1\over{\ln(x)}}dx\,dy$. (Hint: Change the
order of integration.)

%% CORRECT
$e-1$
%% WRONG
$1$
%% WRONG
$-1$
%% WRONG
$e$
%% WRONG
$1/\ln(2)$

%12
%% PROBLEM
Determine which of the following integrals gives the volume of the solid
inside the cylinder $x^2 + y^2 = 2x$ above the $xy$-plane and below the
cone $z=\sqrt{x^2+y^2}$.

%% CORRECT
$\int_{-\pi/2}^{\pi/2}\int_0^{2\cos(\theta)}\int_0^r r\,dz\,dr\,d\theta$
%% WRONG
$\int_{0}^{2\pi}\int_0^{2}\int_0^r r\,dz\,dr\,d\theta$
%% WRONG
$\int_{-\pi/2}^{\pi/2}\int_0^{2\cos(\theta)}\int_0^1 z\,dz\,dr\,d\theta$
%% WRONG
$\int_{-\pi/2}^{\pi/2}\int_0^{2}\int_0^z r\,dz\,dr\,d\theta$
%% WRONG
$\int_{0}^{\pi}\int_0^{2\cos(\theta)}\int_0^r r\,dz\,dr\,d\theta$

%13
%% PROBLEM
Evaluate the integral $\int_1^4\int_0^{1/\sqrt{x}} e^{\sqrt{x}}\,dy\,dx$
by using the change of variables $x = u^2$, $y = v$.
%% CORRECT
$2(e^2-e)$
%% WRONG
$e - 1$
%% WRONG
$2/e$
%% WRONG
$4e^2 - e$
%% WRONG
$e^{\sqrt{3}} - 1$

%14
%% PROBLEM
Let $E$ be the solid bounded by the cylinder $z = 1-x^2$ and the plane
$y = z$ in the first octant.  Determine which of the following integrals
equals $\iiint_E f(x,y,z)\,dV$.

%% CORRECT
$\int_0^1\int_0^{1-x^2}\int_0^z f(x,y,z)\,dy\,dz\,dx$
%% WRONG
$\int_0^1\int_0^{\sqrt{1-z}}\int_0^1 f(x,y,z)\,dz\,dx\,dy$
%% WRONG
$\int_0^1\int_z^1\int_0^{\sqrt{1-z}} f(x,y,z)\,dx\,dy\,dz$
%% WRONG
$\int_0^1\int_0^1\int_0^{1-x^2} f(x,y,z)\,dz\,dx\,dy$
%% WRONG
$\int_0^1\int_0^{1-x^2}\int_z^1 f(x,y,z)\,dy\,dz\,dx$

%15
%% PROBLEM
Let $E$ be the solid between the concentric hemispheres $x^2 + y^2 + z^2 =
16$, $z \ge 0$, and $x^2 + y^2 + z^2 = 4$, $z \ge 0$. Assuming the density
at each point is inversely proportional to its distance from the origin,
find the mass of $E$. (Let $k$ denote the constant of proportionality.)

%% CORRECT
$12k\pi$
%% WRONG
$2k\pi\ln(2)$
%% WRONG
$48k\pi$
%% WRONG
$8k\pi$
%% WRONG
$8k\pi^2$

%16
%% PROBLEM
Let $\C$ be the curve $\r(t) = \langle \sin(3t),\sin(t)+2\sin(2t),
\cos(t)-2\cos(2t)\rangle$, $0 \le t \le 2\pi$. Compute $\int_\C y\,dx +
x\,dy + z\,dz$.

%% CORRECT
0
%% WRONG
1/2
%% WRONG
1
%% WRONG
2
%% WRONG
3/2

%17
%% PROBLEM
Find the equation of the plane tangent to the surface parameterized by
$$\r(u,v) = \langle u - v^2, v - u^2, uv\rangle,\quad-2\le u\le2,\quad-2\le
v\le2$$ at the point $(0,0,1)$.

%% CORRECT
$x + y + z = 1$
%% WRONG
$y+z = 1$
%% WRONG
$x - y + z = 0$
%% WRONG
$x + y - z = 0$
%% WRONG
$-x - y + z = 1$

%18
%% PROBLEM
Suppose the temperature at a point $(x,y,z)$ on the hemisphere $z =
\sqrt{1-x^2-y^2}$ is $T(x,y,z) = 20z^2$. Determine the average temperature
on the hemisphere. The area of the surface is $2\pi$.

%% CORRECT
6.67
%% WRONG
10
%% WRONG
2.09
%% WRONG
15
%% WRONG
12.33

%19
%% PROBLEM
Let $\C$ be the intersection of the cylinders $x^2+y^2 = 1$ and $y^2 + z^2
= 4$ with $z \ge 0$, oriented counter-clockwise when viewed from
above. Evaluate $\int_\C \F\cdot d\r$ where $\F = y\i + 2x\j + e^z\k$.

\centerline{\epsfysize=1in\epsfbox{practice.19.1.eps}
\quad\epsfysize=1in\epsfbox{practice.19.2.eps}}

%% CORRECT
$\pi$
%% WRONG
$2\pi$
%% WRONG
0
%% WRONG
$\sqrt{2} + \pi$
%% WRONG
$\sqrt{2}/2 + 2\pi$

%20
%% PROBLEM
Let $S$ be the sphere $x^2+y^2+z^2 = 1$ and let $\n$ be the outward unit
normal vector to~$S$.  Calculate the flux integral $\iint_S \F\cdot\n\,dS$
where $\F(x,y,z) = xy^2\i + yz^2\j + zx^2\k$.

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

%21
%% PROBLEM
Let  $\F(x,y) = (x + y)\i + (y-x)\j$. Calculate $\int_{\C} \F\cdot d\r$
where $\C$ is the circle $\r(t) = \langle\cos(t),\sin(t)\rangle$, $0 \le t
\le 2\pi$.

%% CORRECT
$-2\pi$
%% WRONG
0
%% WRONG
$3\pi/2$
%% WRONG
$-1$
%% WRONG
$\pi$

%22
%% PROBLEM
Let $\F$ be a vector field defined on $\R^3$. Determine which of the
following conditions guarantees that $\F = \nabla f$ for some function $f$.

%% CORRECT
$\curl\F = \0$
%% WRONG
$\div\F = 0$
%% WRONG
$\div\curl\F = 0$
%% WRONG
$\curl\curl\F = \0$
%% WRONG
$\grad\div\F = \0$

%23
%% PROBLEM
Let $\F= (e^x+yz)\i + (e^y-xz)\j + (\sin(z)+x-y)\k$. Compute $\div\F$.

%% CORRECT
$e^x+e^y+\cos(z)$
%% WRONG
$e^x\i + e^y\j + \cos{z}\k$
%% WRONG
$z-x+1$
%% WRONG
$y+z+\cos{z}$
%% WRONG
$(x-1)\i + (y-1)\j +2z\k$

%24
%% PROBLEM
Determine which of the following integrals gives the area  of the surface
parameterized by $\r(u,v) = u^2\i + uv\j + v^2\k$, $0 \le u \le 1$, $0\le v
\le 2$.

%% CORRECT
$\int_0^2\int_0^1 2\sqrt{u^4 + 4 u^2 v^2 + v^4}\,du\,dv$
%% WRONG
$\int_0^2\int_0^1 \sqrt{u^4 + u^2 v^2 + v^4}\,du\,dv$
%% WRONG
$\int_0^2\int_0^1 u^4 + u^2 v^2 + v^4\,du\,dv$
%% WRONG
$\int_0^2\int_0^1 \sqrt{2}(u^2 + 4u v + v^2)\,du\,dv$
%% WRONG
$\int_0^2\int_0^1 2(u^2 + v^2)\,du\,dv$

%25
%% PROBLEM
Determine which of the following plots is the surface parameterized by
$\r(u,v) = u \cos(v)\i + u\sin(v)\j + v\k$, $-1\le u \le 1$, $0 \le
v \le 2\pi$.

%% CORRECT
\epsfysize=1in \epsfbox{practice.25.1.eps}
%% WRONG
\epsfysize=.75in \epsfbox{practice.25.2.eps}
%% WRONG
\epsfysize=.75in \epsfbox{practice.25.3.eps}
%% WRONG
\epsfysize=.75in \epsfbox{practice.25.4.eps}
%% WRONG
\epsfysize=.75in \epsfbox{practice.25.5.eps}

%% END

\problemspace

\noindent\llap{26.\ }
Let $\F = (e^y + yz)\i + x(e^y + z)\j + (xy - 2)\k$.

\noindent
a) Find a function $f$ such that $\F = \nabla f$.

\noindent
b) Use the Fundamental Theorem for Line Integrals to evaluate $\int_\C
\F\cdot d\r$ where $\C$ is a curve from $(0,0,0)$ to $(3,1,-1)$.

\problemspace

\noindent\llap{27.\  }
Use Green's Theorem to calculate
$$\int_{C} (e^{x^2} + xy) dx + (x + \sin(y^2)) dy$$
where $C$ is the square with vertices $(0,0)$, $(1,0)$,
$(1, 1)$, $(0, 1)$, oriented counter-clockwise.

\bye
