\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, 1997}\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
Compute $\lim_{(x,y)\to(0,0)} {y+xy\over x^2 + xy^2 + y}$
%% WRONG
1
%% WRONG
0
%% WRONG
1/2
%% WRONG
2
%% CORRECT
{\sl does not exist}

%2
%% PROBLEM
Let $\F(x,y,z) = {x\over y}\i + {y\over z}\j + {z\over x}\k$. Compute
$\curl \F$.
%% CORRECT
${y\over z^2}\i+{z\over x^2}\j + {x\over y^2}\k$
%% WRONG
${1\over x}+{1\over y} + {1\over z}$
%% WRONG
${-y\i+z\j-x\k\over x^2+y^2+z^2}$
%% WRONG
${x^3z^2 + x^2y^3+ y^2z^3\over x^2y^2z^2}$
%% WRONG
${1\over x}\i+{1\over y}\j + {1\over z}\k$


%3
%% PROBLEM
Compute the angle between the planes $2x+y+z = 1$ and $x+2y-z=1$.
%% CORRECT
$\pi/3$
%% WRONG
$0$
%% WRONG
$\pi/2$
%% WRONG
$\pi/4$
%% WRONG
$3\pi/4$

%4
%% PROBLEM
Determine the equation of the plane through the point $(3,0,1)$ that is
parallel to the vectors $\i+\j - \k$ and $\i-\j$.

%% CORRECT
$x + y + 2z = 5$
%% WRONG
$x + y - z = 2$
%% WRONG
$x - y = 3$
%% WRONG
$x + 2y + z = 4$
%% WRONG
$x - y - 2z = 1$

%5
%% PROBLEM
Suppose a particle's position is given by $r(t) = e^{t^2-1}\i +
e^{1-t^2}\j + t \k$. Determine the particle's speed at the point
$(1,1,-1)$.
%% CORRECT
3
%% WRONG
$2\i-2\j + \k$
%% WRONG
$\k$
%% WRONG
$\sqrt{4e^2 + 4e^{-2} + 1}$
%% WRONG
1

%6
%% PROBLEM
Consider the curve parameterized by $r(t) = {t^2\over 2}\i + t\j - t\k$.
Calculate the unit normal vector to this curve at the point
$(1,\sqrt{2},-\sqrt{2})$.
%% CORRECT
${1\over\sqrt{2}}\i - {1\over 2}\j + {1\over 2}\k$
%% WRONG
${1\over\sqrt{2}}\i + {1\over 2}\j - {1\over 2}\k$
%% WRONG
${1\over\sqrt{2}}\i + {1\over 2}\j + {1\over 2}\k$
%% WRONG
${1\over\sqrt{2}}\i - {1\over 2}\j - {1\over 2}\k$
%% WRONG
${-1\over\sqrt{2}}\i - {1\over 2}\j + {1\over 2}\k$


%7
%% PROBLEM
Let $f(x,y,z) = {x^2 - y z\over x y^2}$. Compute $f_{xyz}$
%% CORRECT
${-1\over x^2y^2}$
%% WRONG
${-2\over x^2y^3}$
%% WRONG
${-2\over x^3y^2}$
%% WRONG
${-4\over x^3y^3}$
%% WRONG
${-1\over xy}$

%8
%% PROBLEM
If $z = \cos(x^2 - y^2)$ and $x = e^u$, $y = u e^{-v}$, determine which of
the following expressions gives $\d{z}/du$.

%% CORRECT
$-2(xe^{u} - ye^{-v}) \sin (x^2 - y^2)$
%% WRONG
$2(e^{2u} - e^{-2v})\sin(e^{2u} - e^{-2v}u^2)$
%% WRONG
$-2u^2e^{-2v}\sin(x^2-y^2)$
%% WRONG
$-2e^{2u}\sin(x^2-y^2)$
%% WRONG
$-2(x-y)\sin(x^2-y^2)$

%9
%% PROBLEM
Calculate the derivative of the function $f(x,y,z) = xy^2 - z^3$ at
$(-1,2,1)$ in the direction of the vector $(2,1,2)$.
%% CORRECT
$-2/3$
%% WRONG
$1/3$
%% WRONG
$-2$
%% WRONG
1
%% WRONG
0


%10
%% PROBLEM
Determine the equation of the plane tangent to the surface $z^2 = x^3 +
y^3$ at the point $(1,2,3)$.

%% CORRECT
$x + 4y - 2z = 3$
%% WRONG
$(x-1) + 2(y-4) + 3(z+2) = 0$
%% WRONG
$(x-1) + 4(y-2) = 0$
%% WRONG
$x + 2y = 0$
%% WRONG
$3x + 4y - 6z = 9$ 


%11
%% PROBLEM
Determine which of the following statements describes the function
$$f(x,y) = y^2 (x-1) + x^2$$

%% CORRECT
$f(x,y)$ has a saddle point at $(0,0)$.
%% WRONG
$f(x,y)$ has a local minimum at $(0,0)$.
%% WRONG
$f(x,y)$ has a local maximum at $(0,0)$.
%% WRONG
$f(x,y)$ has a critical point at $(1,\sqrt{2})$.
%% WRONG
$f(x,y)$ does not have a critical point at $(0,0)$.


%12
%% PROBLEM
Find the maximum of the function $f(x,y) = x(y+1)$ subject to the
constraint\hfil\break
$2 x^2 + y^2 = 1$.

%% CORRECT
0.919
%% WRONG
1.431
%% WRONG
1.564
%% WRONG
1.111
%% WRONG
1.333

%13
%% PROBLEM
Find all the critical points of the function $f(x,y) = x^3y - xy + 3x^2 + 2x$.
%% CORRECT
$(0,2)$, $(1,-4)$, $(-1,2)$
%% WRONG
$(0,2)$, $(1,-4)$, $(1,4)$, $(-1,2)$
%% WRONG
$(0,2)$, $(1,-4)$, $(-1,2)$, $(1,2)$
%% WRONG
$(0,2)$, $(-1,4)$, $(1,-2)$
%% WRONG
$(0,2)$, $(-1,-4)$, $(1,-4)$, $(-1,2)$, $(1,2)$

%14
%% PROBLEM
Reverse the order of integration in the integral
$\int_{1}^{2}\int_{\sqrt{x-1}}^{\root3\of{x-1}} f(x,y)\,dy\,dx$.

%% CORRECT
$\int_0^1\int_{y^3+1}^{y^2+1} f(x,y)\,dx\,dy$
%% WRONG
$\int_{\sqrt{x-1}}^{\root3\of{x-1}}\int_{1}^{2} f(x,y)\,dx\,dy$.
%% WRONG
$\int_1^2\int_{y^2+1}^{y^3+1} f(x,y)\,dx\,dy$
%% WRONG
$\int_0^1\int_{1+\root3\of{y}}^{1+\sqrt{y}} f(x,y)\,dx\,dy$
%% WRONG
$\int_0^2\int_{y^2}^{y^3} f(x,y)\,dx\,dy$



%15
%% PROBLEM
Find the area of the region below the curve
$r=\cos(\theta)-\sin(\theta)$ and above the $x$-axis.

\centerline{\epsfxsize=2.5truein\epsfbox{final.15.eps}}

%% CORRECT
${\pi-2\over 8}$
%% WRONG
${\pi-1\over 4}$
%% WRONG
${3\pi-2\over 4}$
%% WRONG
${\pi-3\over 4}$
%% WRONG
${2\pi-3\over 8}$


%16
%% PROBLEM
Find the average height above the $xy$-plane of the points in the solid
region of the first octant below the inverted cone $z = 2 - \sqrt{x^2+y^2}$.
The volume of the solid is $2\pi/3$.

\centerline{\epsfxsize=1.75truein\epsfbox{final.16.eps}}

%% CORRECT
$1/2$
%% WRONG
$2/3$
%% WRONG
$\pi/2$
%% WRONG
$\pi/3$
%% WRONG
$7/8$


%17
%% PROBLEM
Let $D$ be the solid region between the spheres of radii 1 and 2. Compute
the integral $\iiint_D z^2\, dV$.

%% CORRECT
${124\pi\over 15}$
%% WRONG
${12\pi\over 5}$
%% WRONG
$24\pi$
%% WRONG
${8\pi\over 3}$
%% WRONG
$4\pi$


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

%% CORRECT
$-2uv$
%% WRONG
${1\over 2uv}$
%% WRONG
${1\over 4\sqrt{x^2-y^2}}$
%% WRONG
$-{1\over  2\sqrt{x+y}}$
%% WRONG
${1\over 4}$

%19
%% PROBLEM
Let $\CC$ be the semi-circle $y = \sqrt{1-x^2}$ and let $\T$ be the
unit tangent vector to $\CC$ oriented counterclockwise. Determine
which of the following integrals gives the value of $\int_\CC
\F\cdot\T\, ds$ where $\F(x,y) = y^2\i + x^2\j$.

%% CORRECT
$\int_0^{\pi} \cos^3(t) - \sin^3(t) \, dt$
%% WRONG
$-\int_0^{\pi}\cos(t)\,dt$
%% WRONG
$\int_0^{\pi} \cos(t) + \sin(t) \, dt$
%% WRONG
$\int_0^{\pi} \sin^2(t) - \cos^2(t) \, dt$
%% WRONG
$\int_0^{\pi} dt$

%20
%% PROBLEM
Compute the line integral $\int_\CC y\, ds$ where $\CC$ is the curve
parameterized by $\r(t) = t^4\i - t^2\j$, $0\le t \le 1$.

%% CORRECT
${1\over 12}(5^{3/2} - 1)$
%% WRONG
${1\over 6}(3^{1/2} - 1)$
%% WRONG
${1\over 3}(2^{3/2} - 1)$
%% WRONG
${1\over 4}(3^{5/2} - 1)$
%% WRONG
${1\over 2}(6^{1/2} - 1)$


%21
%% PROBLEM
Let $\CC$ be the curve parameterized by  $\r(t) = \cos(3t)\i +
\sin(2t)\j + t\k$, $0\le t \le 2\pi$. Calculate $\int_\CC y e^{xy}\,dx +
x e^{xy}\,dy + \,dz$.

\centerline{\epsfxsize=1.75truein\epsfbox{final.21.eps}}

%% CORRECT
$2\pi$
%% WRONG
$1$
%% WRONG
$0$
%% WRONG
$e-2\pi$
%% WRONG
$e$

%22
%% PROBLEM
Determine which of the following integrals gives the area of the
surface $z^4 = 4(x^2 + y^2)$, $0 \le z \le \sqrt{2}$.

\centerline{\epsfxsize=2.5truein\epsfbox{final.22.eps}}

%% CORRECT
$\int_0^{2\pi}\int_0^1 \sqrt{r^2 + r/2}\,dr\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^1 4r\sqrt{4r^2 + z^2}\,dr\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^1 8\sqrt{2}r\,dr\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^1 4r^2\sqrt{2}\,dr\,d\theta$
%% WRONG
$\int_0^{2\pi}\int_0^1 4r^2\sqrt{4r^2 + r}\,dr\,d\theta$


%23
%% PROBLEM
Let $\Sigma$ be the portion of the paraboloid $z = x^2 + y^2$ in the
first octant below $z=2$ and let $\n$ be the unit downward normal
vector to $\Sigma$.  Compute the flux integral
$$\iint_\Sigma (y\i - x\j + \k)\cdot\n\,d\sigma$$

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

%% CORRECT
$-{\pi\over2}$
%% WRONG
${4\pi\over3}$
%% WRONG
0
%% WRONG
$4\pi$
%% WRONG
$-{5\pi\over2}$

%24
%% PROBLEM
Let $\Sigma$ be the portion of the parabolic sheet $z=1-x^2$ that lies inside
the square\hfil\break
$-1\le x \le 1$, $-1\le y \le 1$, and let $\CC$ be the boundary
of $\Sigma$ oriented counterclockwise when viewed from above.
Use Stokes' Theorem to rewrite the line integral $\int_\CC (y\i + z\j +
x\k)\cdot d\r$ as a double integral.

\centerline{\epsfxsize=2.5truein\epsfbox{final.24.eps}}

%% CORRECT
$-\int_{-1}^{1}\int_{-1}^{1}(2x+1)\,dy\,dx$
%% WRONG
$\int_{-1}^{1}\int_{-1}^{1}(\i+\j+\k)\,dy\,dx$
%% WRONG
$\int_{-1}^{1}\int_{-1}^{1}(x+y+x^2)\,dy\,dx$
%% WRONG
$-\int_{-1}^{1}\int_{-1}^{1}(2xy + x)\,dy\,dx$
%% WRONG
$\int_{-1}^{1}\int_{-1}^{1}(y\i + z\j +x\k)\sqrt{4x^2+1}\,dy\,dx$

%25
%% PROBLEM
Let $D$ be the solid cylinder defined by $x^2+y^2 \le 4$, $0\le z \le
3$.  Let $\Sigma$ be the boundary of this solid and let $\n$ be its
outward unit normal vector. Use the Divergence Theorem to compute the
flux integral $\iint_\Sigma(xy\i - y^2\j + z^2\k)\cdot\n\,d\sigma$.

%% CORRECT
$36\pi$
%% WRONG
$18\pi$
%% WRONG
$12\pi$
%% WRONG
$9\pi$
%% WRONG
$6\pi$

%% END
\end

%% END
\bye
