\documentclass{article}
\title{Exams from Calculus 223}
\author{J. Robbin }
\date{Fall 1994}
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\newcommand{\p}{\partial}

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\begin{document}

\maketitle

\begin{quote}
\em
Warning: Some of this material may not be included
in Math 234 for 2001.
\end{quote}

\bigskip

\centerline{\Large Calculus 223 First Exam  \\ Thursday September 29, 1994}

\bigskip

\noindent{I} Find an equation for the tangent plane to the surface
$$
  x^2-y^2+z^2=4
$$
at the point $(2,-3,3)$.

\bigskip

\noindent{II} (a) Find the unique critical point of the function
$$
     f(x,y) = x^2+ 3xy + 2y^2 -8x-11y + 30.
$$
\medskip

\noindent(b) Is this critical point a minimum, maximum, or saddle?

\medskip

\noindent(c) Does the function $f(x,y)$ take negative values?
(I.e. is there a point $(x,y)$ where $f(x,y)<0$?).
{\em Justify your answer.}

\bigskip

\noindent{III} Find
$\ds \left(\frac{\p w}{\p y}\right)_x$  at
$(w,x,y,z)=(0,1,2,3)$ if
$$
4x+5y+6z=32+w \qquad
\mbox{ and }
\qquad
7x^2+8y^2+9z^2=w+120e^w.
$$
\bigskip

\noindent{IV} Find the polynomial of degree $2$ which best approximates
$$
f(x,y)=  \sin(xy)
$$
near $(x,y)=(1,\pi)$.

\bigskip

\noindent{V} Find the absolute maximum and the absolute minimum of
$$
        f(x,y) = (x-1)(y-2)
$$
in the closed
triangle $0\le x$, $0\le y$, $x+y\le 7$
bounded by the $x$-axis, the $y$-axis, and the line
$x+y=7$.

\bigskip


\noindent{VI} Find the  point on the ellipse
$2x^2+3y^2=11$ where the function  $f(x,y)=8x-6y$
achieves its maximum.

\bigskip

 \noindent{VI}\footnote{This problem was not included on the exam.}
 Let $T=f(x,y)$ be the temperature at
 the point $(x,y)$ on the circle
 $$
   x= \cos\theta,\qquad y=\sin\theta,
 $$
 and suppose that
 $$
      \frac{\p T}{\p x} = 2x-y, \qquad
      \frac{\p T}{\p y} = 2y-x.
 $$
 Find where the maximum temperature on the circle
 occurs. {\em Be sure not to specify where the maximum
 occurs and not the minimum. Be sure to justify your answer.}


\bigskip

\hrule

\medskip

\centerline{\Large Calculus 223 Second Exam  \\ Tuesday November 1, 1994}

\bigskip

\noindent{I} Evaluate
$
\ds \int_{x=0}^1 \int_{y=2x}^1 \int_{z=x^3+y}^{x^2+2y}  y\, dz\, dy\, dx.
$

\bigskip

\noindent{II} The force at the point $(x,y)$ is
$$
   {\bf F}(x,y) = x^2y{\bf i} + 2xy^2{\bf j}.
$$
Find the work
$$
     W = \int_C {\bf F}\cdot{\bf T}\, ds
$$
done in moving a particle from $(0,0)$
to $(2,4)$ along the curve $y=x^2$.
%
% Make sure students have been assigned a problem like this one.
%

\bigskip

\noindent{III} Evaluate the integral
$\ds\int\!\int_R x\,dx\,dy$ where
$R$ is the triangle with vertices
$(1,2)$, $(3,3)$, $(4,5)$.

\bigskip

\noindent{IV} The transformation $(x,y)=T(u,v)$ is given by
$$
     x = 1+ 2v+u^2,\qquad y = \frac{u}{3}.
$$
It transforms the triangle
$$
      G:\qquad  0\le u \le v \le 1
$$
in $(u,v)$ space to a region $R$ in $(x,y)$ space.
The region $R$ is bounded by three curves.

\medskip




\noindent(a) Sketch the region $R$ on the axes provided.
Indicate clearly which points of the boundary of $R$
correspond to the vertices $(0,0)$, $(0,1)$, $(1,1)$
of the triangle $G$.
Find equations in $(x,y)$ for each of the boundary curves.
(Indicate clearly which boundary curve of $R$ corresponds to
which equation.)

\medskip

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 \put(1,5){\line(1,0){4}}
\put(6,6){$G$}
\put(20,0){\vector(1,0){28}} \put(50,0){$x$}
\put(23,-5){\vector(0,1){18}} \put(23,14){$y$}

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% \put(10,-1.9){\line(1,0){5.8}}
\put(13,3){$T$}

\end{picture}

\medskip

\noindent(b) Evaluate $\ds \int\!\int_R (x+y^2)\,dx\,dy$.

\bigskip


\noindent{V} Let $C$ be the cardioid with polar equation
\message{Draw PICTURE!}
$$
      r = 1-\cos\theta, \qquad 0\le\theta\le 2\pi.
$$

\medskip

\noindent(a) Evaluate the integral
$$
\oint_C x\,dy
$$
using the definition of line integral.
You may leave your answer in the form
of a single definite integral.

\bigskip

\noindent(b) Evaluate the integral
using Green's theorem. A numerical answer is required here.

\bigskip


\noindent{VI}
The gravitational potential at the point
$(0,0,z_0)$ due to a uniform mass distribution
in the spherical shell
$$
   R:\qquad a\le \sqrt{x^2+y^2+z^2} \le b
$$
is given by the triple integral
$$
    \int\!\int\!\int_R \frac{dx\,dy\,dz}{f(x,y,z)}
$$
where $f(x,y,z)$ is the distance from the point $(x,y,z)$
to the point $(0,0,z_0)$.

\medskip

\noindent(a)  Show that $f(x,y,z) = \sqrt{\rho^2+z_0^2-2\rho z_0\cos\phi}$.

\medskip

\noindent(b) Evaluate the potential as a function of $z_0$.
Hint: $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$.
{\em Leave your answer as a definite integral in $\rho$.
You will be asked to evaluate this integral on the next page.}

\medskip

\noindent(c) Evaluate the integral in~(b) when $z_0>b$

\medskip


\noindent(d) Evaluate the integral in~(b) when $0<z_0<a$.

\bigskip

\hrule

\medskip

\centerline{\Large Some questions from the Final 223 Exam in 1994}

\bigskip

\noindent{I} Find an equation for the tangent plane
to the surface $z=1+x^2+y^3$ at the point
$(x,y,z)=(2,1,6)$.


\bigskip

\noindent{II} Find a function $w=f(x,y)$ whose first partials are
$$
   \frac{\p w}{\p x} = 1+e^x\cos y, \qquad
    \frac{\p w}{\p y} = 2y -e^x\sin y
$$
or prove that there is no such function.



\bigskip



\noindent{III} Let $C$ be the triangle with vertices
$(1,2)$, $(3,2)$, $(2,5)$.
Evaluate the line integrals
$\oint_C x\,dx -y\,dy$ and
$\oint_C y\,dx-x\,dy$.
Both integrals are to be traversed in the counterclockwise direction.


\bigskip

\noindent{IV} The transformation $(x,y)=T(u,v)$ is defined by
$$
      x = e^{2u}+e^v, \qquad y = e^u+e^v
$$
carries the unit square
$0\le u\le 1$,   $0\le v\le 1$
in the $(u,v)$ plane to a region $R$ in the $(x,y)$ plane
shown in the diagram.

\smallskip

\noindent(1) Complete the table to give the
coordinates  of the vertices of $R$.

\message{DRAW R}



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\put(0,0){\input{graph}}
\put(30,10){
\begin{tabular}{l|l} $P$ & $\qquad (x,y)\qquad $\\ \hline\hline
   $A$ &   \\ \hline
   $B$ &   \\ \hline
   $C$ &   \\ \hline
   $D$ &   \\ \hline
\end{tabular}
}
\end{picture}

\medskip

\noindent(2) Which of the four sides of $R$ are straight line segments?
(Circle one.)

\medskip

\noindent \hfill  none
\hfill  all \hfill $AB$ and $CD$ \hfill $AC$ and $BD$ \hfill

\medskip

\noindent(3) Find the area of $R$.

\bigskip


\centerline{\Large A question on the Divergence Theorem}


\bigskip

Let $W=\rho^{-1}$ where $\rho=\sqrt{x^2+y^2+z^2}$.

\medskip

\noindent(1) Calculate the gradient $\nabla W$ and the divergence of the gradient
$\nabla\cdot\nabla W$,

\bigskip

\noindent(2) Calculate the outward flux
$$
     \int\!\!\!\!\int_S \bfF\cdot\bfn \,d\sigma
$$
over the sphere $\rho=h$  of radius $h$.
Here $d\sigma$ denotes the area element on the sphere
and $\bfn$ denotes the outward unit normal to the sphere.

\bigskip

\noindent(3) Calculate the outward flux
$$
     \int\!\!\!\!\int_S \bfF\cdot\bfn \,d\sigma
$$
over the ellipsoid $S$ defined by
$$
\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1.
$$
Here $d\sigma$ denotes the area element on the ellipsoid
and $\bfn$ denotes the outward unit normal to the ellipsoid.

\end{document}
